St Andrews Lynx

Don't think of it as a test for you ("how many insightful comments can I make about the papers they've written?!"). It's a way to figure out what the postdoc is like as a person and if they'd be a good personality fit for the group.
I'd do some background reading on the person's CV (or LinkedIn profile, or website, whatever is available). Find out where they did their undergrad/PhD, and if there is anything interesting in their professional background you could ask them about ("I saw you did an internship at [pharma company] for 6 months after you graduated - what was that like?" "I saw you presented at X conference a few years back - I was thinking about attending that conference next summer, would you recommend it?").
Ask them about their hobbies. Did they see any good movies lately. What sports do they follow. Show interest in their responses. Ask follow-up questions.
Topics to avoid include anything political or religious. Do not assume anything about the candidates' marital/family status/sexual orientation - unless they bring up children or a spouse in the conversation I wouldn't ask them about that. 
Hopefully the postdoc should have questions about general life in the area as well as questions about how the group operates. 
 

As another point. Encourage A and D to make use of the university counselling services to talk through their feelings and deal with what has happened (in addition to doing what rising_star has outlined). It sounds like everybody involved is suffering from a degree of trauma - give A the time and space to process their feelings and come to their own decisions. If A decides she wants to keep silent and pretend like the whole thing never happened...that is her right to do so, even if you'd rather she didn't. 
The DGS and Dept Chair are there for issues related to the program (conflict with your advisor, requirements for completion, etc). An event that happened outwit the Dept is beyond their remit. It is also likely that they won't have had special training in how to deal with sexual assaults, nor are they bound to confidentiality. So even if those figures want to help...they may do/say the wrong things. 
C is somebody you are better off staying away from. You are unlikely to get closure or any kind of admission of guilt/apology from her. Don't interact with her beyond what is required in the context of your professional work. 

In my department, we generally keep the bulk of the conversation at lunch and dinner away from the visitor's research. This is because we often also have 30 minute time slots for one-on-one or small group meetings, which are generally all about research. And, usually the visitor has back to back meetings scheduled during the entire day where all they will talk about is their research.
I'm not saying that we can never talk about research at a lunch/dinner, but generally the meal time is supposed to be a break time for the visitor and also a chance for grad students to learn some other insight. So, we would often ask about things like "How did you decide that you want to be a postdoc?", "Do you have any tips for the job market in field X?", "What is it like to live in [their city/their university/etc]". Or maybe get advice on fellowships for grad students, etc. I'd say that the meals, in my department, are more for us to know the visitor as a person rather than as a researcher.
Additional tip: I find that one of the easiest ways to open a conversation is to ask about their travel to your city and like kimmibeans suggested, ask them "is it your first time in [city]?" etc. Usually once the conversation starts to flow, it will naturally find interesting topics

The good news is that there will be a nice chunk of grad students in your cohort who are ALL starting out at the same time, and many of whom are completely new to the area. If you get chatting to them during the Orientation Week (where there are usually a lot of socials) and follow-up in the coming weeks, then I think you'll form friends naturally. 
 
What I like about grad school is that you don't have to stay up to 3am partying as a pre-requisite to making friends - unlike in undergrad and the Greek Life spheres. Grad students are typically the ones who DIDN'T go to the frat parties and WEREN'T the coolest undergrads (although many are really cool, of course).
 
My best advice is to be proactive in making friends. Go up and talk to new folk. Be the one who suggests doing something on the weekend. Invite yourself to activities by asking if you can come along. Give strangers a second chance (first impressions aren't always the most accurate).  

A lot of the questions you post here, including this one, cannot be resolved by anonymous members on an internet forum. People here give excellent general advice, but they cannot advise you on issues that by definition must be resolved between you and the people responsible for those issues. We don't know what the policy on student status is at your university. You being an international student further complicates matters, because then we not only have to guess the policy of the Registrar at this unnamed university, but also the International Office. Those are the entities that can give you the answers you want, not us. You should go talk to them.

Doesn't sound completely "out there". The conference expenses at most places are paid on a reimbursement basis. 
Things like badly-run core PhD classes and poorly-attended seminars can be found at other schools too, in varying degrees. In general I don't think that grad schools put much effort into classes for the PhD students - at some level you're expected to have done most of your learning as an undergrad, and the classes are more about ensuring that everyone in the program is up to the same level. 
The administrative issues (payment delays, not communicating PhD program requirements promptly) are much more of an issue. In a good dept, the admin will hum along in the background and be so smooth you hardly notice it's there. Never under appreciate a good admin team! 
***
My advice? Stick with the school and focus on your research.* Classes and seminars don't matter - the most important thing left after your PhD will be your research and relationship with your lab mates & PIs. If the admin are flaky then it is important to be proactive and read all the documents/websites about PhD requirements at a departmental and university level. If you don't have a union then I wouldn't waste your energy on changing things in a Dept that no one has an incentive to change. You're here to do research, not get dragged into battles you can't win. 

Oh man. Where is your advisor in all this? Do they know about your philosophy on grad school/academia? You need to talk to them (or some other senior person you trust) about your feelings, because you're going about this all wrong.
Yes, many people in academia share your enthusiasm for pursuing knowledge. So do many butchers, bakers, and candlestick makers. But you are not paid to consume knowledge; you are paid to produce knowledge. You have to concentrate on one thing not because it is a mindset foisted on you by evil neoliberal oppressive academia, but because if you work in a niche, you will be able to learn all or almost all there is to know about that niche, and you will be more efficient at producing knowledge other people will find useful. That's what gets you hired into an academic job - not how many classes you take, or how many irrelevant factoids you know. You are not a student; you are a future professional researcher. Grad school isn't a free opportunity for you to chill out and pursue your personal hobbies in CS or statistics - it is borrowed time where you are free of practical obligations such as having a full-time job or finding a way to work with experts in your field and have access to a scholarly library, which you are supposed to use to show the professional community that you can produce competent research and can be considered for a similar, more permanent, and better paid job doing what you have proved you do best. 
Your research and teaching are the only things that matter. Teaching, because that's what pays your bills, and that's what's likely to pay your bills in the future barring a major, major stroke of luck. Research, because that is what you are here to do. They are the two pillars that support your career. If you enjoy pursuing knowledge for the sake of knowledge and taking classes in a variety of subjects for the hell of it, but don't enjoy teaching and aren't dedicated to your research, you should not have pursued academia.
Your advisor needs to know about this because I think you need help figuring out your priorities right now. It may seem counterintuitive, but the wrong priorities, the wrong attitude, and a lack of understanding of how this system functions are probably (can't say definitely because I don't have generalizable evidence) the leading cause of brilliant people not getting academic jobs.

does the gym have a bed and a kitchen?

Hopefully your changes in sleep patterns are a result of something more benign, such as stress.
I don't think there is a way to improve lab safety culture without a bit of confrontation. Standing up in group meeting and telling everybody "Be safer" doesn't make much of a difference if people (i) don't think they are to blame (ii) don't care about the consequences of not following safety procedures (iii) don't know how to work safer. If there are 1 or 2 offenders then I would talk to them in person and in private to explain what the problem with their practices are, and - this is crucial - why it is a problem. A "demonstration of good safety practices" for the compounds in question might be an option if there are a group of people using the particular chemicals.
Give your PI a private heads-up that there are safety non-compliance issues - you don't have to name names to the boss, but they should know that there is a problem. 
Think about changes you can make to lab set-up to minimise exposure. Can the diamond saw be moved to a fume hood? Can the precusor(s) be stored in the glovebox? 
Also, you can try contacting your university's health & safety department. At our university, they can give labs/students advice on best practices for using/handling/storing particular chemicals, and they can also come around to check out the lab set-up (not in the context of a safety audit) for the chemical. 
 
 

A couple things:
I'm so sorry about your mother. I wish her all the best, and you, too.
You can take leave while on an NSF - it's called Medical Deferral status. If an immediate family member (including a parent) experiences a serious illness, you can be granted a Medical Deferral. You can then use the deferred months of your fellowship later on. You have to contact the administrator (coordinating official) of the NSF on your campus and get an approved medical leave from your university, and then submit a request for medical deferral. If you look at the NSF Administrative Guide it's all laid out there. (http://www.nsf.gov/pubs/2011/nsf11031/nsf11031.pdf). This sounds like exactly the kind of situation a medical deferral was invented for.
You may need to spend some time thinking about whether your anxiety and depression is due to PhD studies in general, to your specific program, or to the very difficult period you are going through in your life. That should help you determine what to do here. A leave of absence is a really good time to do that, even if you are spending most of it caring for your mother and getting treatment of your own. First of all, I think regardless of whether you stay or go, put yourself and your family first. You need to care for your mom and you need to care for yourself. That seems to signal getting out quick and getting back home to the Midwest. You aren't sacrificing your career opportunities. You are clearly a bright, intelligent, driven person who managed to get two independent forms of funding AND into a PhD program. You'll be able to achieve in your career. If your program is so terrible as to not be understanding during this very stressful time in your life, forget about them - it may be best to move on. Either way, focus on taking the time to care for your mom if you know in your heart that's what you should do.
But after you get that squared away, then the idea is to reflect some and think about whether it's your department or the program that's stressing you out. Before your mom was diagnosed with breast cancer, how were you doing? Were you enjoying your work and were your professors responding to you in a different way than they are now? Or were you unhappy from the start? Think about the sources of your happiness, and think about your work. Imagine if you were in a lab with your favorite professor at this program, and they were supportive and you were doing interesting research. Would you still love graduate school? Would you still want a PhD? These questions are difficult to answer; I had a difficult time answering them myself until after I had finished my PhD.
One recurring thing I see in your post is that your own perception of your work doesn't seem to match up with outside perceptions of your work. For example, three separate professors have said that they don't want to take you on in their lab because it appears that your interests lie elsewhere (which is a nice way of saying they don't feel you were engaged enough in the lab) and/or because it appears you need more mentorship (which is a nice way of saying that they don't think you are independent enough ). The fact that they don't even want to take you on even though they don't have to pay a dime for you is very telling - but telling of what is the question. Is it just that your mom's illness affected your work in ways that you didn't see or realize from the outside? Or is there a kernel of truth independent of your mom's illness? You say you don't understand their comments. Have you asked them what they meant or for concrete examples of this? What about the PhD student who advised you not to join her lab? Was her advice given in a constructive way or a petty way? Are you comfortable enough with her to go back and have a chat about what she thinks you need to improve? Is your mind open enough to hear some constructive criticism about yourself?
Quite frankly, grit and determination are not enough. 16-hour days in the lab sounds very stressful, and if you were doing this for days on end while also dealing with your mom's sickness and traveling back and forth, the quality of your work may not have held up. Also - different labs are different, but in most labs you shouldn't have to work 16 hours in the lab to be productive on a regular basis (maybe occasionally, but not forever). Days longer than about 10 hours really aren't sustainable by humans for long periods of time, as studies show that really we can only do about 6-8 hours of productive work before our brain starts switching to other things and we lose productivity. When you're in the lab for a long time, are your days really productive or are you actually spending several of those hours procrastinating? And besides all that, what is the quality of the work that you turn out? 12 hours in the lab won't be important to your PIs if you don't have good work to show for it. I'm not saying you do or don't - I'm just saying examine these as potential reasons they don't want you in their labs.
Another possibility, of course, is that this is the kind of program that expects you to sacrifice all personal and family interests in pursuit of Science, and that these professors are part of a culture that embraces that - so they are avoiding you in the lab not because of your work but because you have a sick mom and they don't want a PhD student who has a commitment to caring for her sick mom. It's unlikely this is the case, but it's still a possibility, so examine that as well.
 

If your funding doesn't require that you TA, DO NOT TA, focus on doing well in your classes and especially your research. You can always TA later.

First step - find out if you can take medical leave on the NSF grant. The admin folk in your Dept will know (you don't need to go into the details of why you need to take the leave with them, and they shouldn't even ask). Sure it would complicate things, but apartments can be sub-let and I think family/your mental health comes first.
Transferring sounds like a valid option, but I also want to point out that (i) you don't need a PhD to get a good job (ii) there's nothing wrong with quitting a PhD program if it's a bad fit for you, or if other life stuff happens. Sometimes, walking away is the most courageous thing you can do. Also, the more you can understand about why things went wrong in this program, the better-prepared you will be for any future programs and labs. If it was just a case of the Dept being toxic, then you can learn how to better screen for toxic PIs and Depts. If there was an issue of fit between you and those labs, then you can learn more about the best type of lab environment for you and how to tell what it is a PI is looking for in their rotation students. 

I'll be quick because I have to go run a errand, while the conversation is sort of interesting, from a legal perceptive its a non starter. The first amendment does not say you can't be punished for your speech, it does say that congress cannot make laws " bridging the freedom of speech ". Colleges are not congress and they don't make laws, the first amendment doesn't really apply here. State constitutions might have something to say but that will depend on the state. 

Two points: First, almost all universities receive federal funds, including private universities (through grants etc.) and therefore are subject to federal laws. Schools will lose their funding if they do not follow the law, and you are right about the importance of schools following the law.
Second, I don't think you have the right definition of free speech. Free speech means you can say what you want without legal repercussions but there can still be other consequences. If you apply to my department and in the interview, you tell us that the moon is made out of cheese and that all of the scientific studies on the moon is all wrong and that nothing we say can ever convince you that the moon is not made of cheese, then you will not be accepted. Your right to free speech is still protected: we do not physically stop you or restrain you from saying your belief on the moon. We do not arrest you or compel the government to throw you in jail for this speech. However, this belief (specifically, the refusal to consider future evidence and modify beliefs) is incompatible with the scientific process and there is no way you would be successful as a graduate student in our field. It shows that you are not properly prepared for scientific research and therefore, we will not accept you into the program. This is not a violation of your free speech rights.
When I first started replying, I wasn't sure if people were meaning examples like the one I just gave, or examples like a school only accepting people from a certain religion. Note that free speech is not the only law that schools have to follow. For example, Title IX does not allow schools to discriminate on the basis of sex, requires fair treatment for pregnant students, requires equal opportunities for all students, and requires the school to protect students from sexual harassment and bullying. Therefore, if you are at a school, a faculty member who says something like "I think women aren't good scientists and so I never accept any of them into my lab" will face consequences** for these words. These actions are not protected by freedom of speech. 
(**Note: I'm not saying that they will be fired, as the consequence needs to be appropriate to what they did. It could be a range of things from education to dismissal from the university).

All of these things are bad things that should not be going on at an interview. Faculty and staff at public and private universities should not be able to ask these questions or make admissions decisions based on these factors. When the term "good fit" is used in the context of admissions though, most people do not mean good fit as in a compatible religious or political belief. I think there should be consequences for schools that engage in this bad admissions practice, but I don't think preventing interviews at all is a good solution. 

I don't think interviews are a violation of free speech rights. An applicant's free speech rights are not violated if they are denied a position based on what they say. Freedom of speech is not the same as freedom of all repercussions of said speech. 
I do agree with you that universities shouldn't reject/admit someone because of things related to freedom of speech like agreeing with certain political views, religious beliefs, or other opinions. However, this is not what reasonable people mean when they say graduate admissions are looking for a "good fit". When we say "good fit", we usually mean "fit" in terms of things that are relevant to the work. For example, will the school's resources and the applicant's goals match up? It doesn't make sense to admit a student who wants to use telescope data to a school that has no telescope data.
It may be the case that some schools are using the guise of "good fit" to discriminate against applicants for their opinions/beliefs/etc. not relevant to their work. This would be very bad. There are laws against hiring discrimination (doesn't matter whether the money is federally sourced or not) that are meant to protect employees and these laws restrict the types of questions that could be asked in an interview. I know that grad student applicants aren't necessary in the same category as employees and may not always be fully protected by these laws, but there are mechanisms that are there to try to prevent schools from doing this.
((In this discussion, I am assuming you are worried about interviews being used to unfairly reject candidates for non-relevant reasons such as political or religious beliefs. If this is not what you mean, would you mind clarifying?))

Are they also going to eliminate interviews for public-sector work positions? Interviews are about more than just "good fit"; they allow the hiring/admissions committee to assess competency, ability to think on the spot, verbal and social communication skills, much more, and yes -- fit between the program/employer and potential student/employee. 
Edit to add: Also, I think interviews can be just as beneficial for the students/person being hired as for the school/employer. I've withdrawn my name from consideration for work positions after doing an interview when I got a weird vibe or felt like I didn't really mesh with their personalities or the work responsibilities. Other times I've gone from feeling so-so about a position to being really hopeful and excited after an interview. Some things can't be judged by GPA, test scores, and 3rd party reference letters.

Well, on the one hand, it seems none of the universities are a very good research fit. I can't, in good conscience, recommend anyone going to a PhD program where they only have a single PI/research group they want to work with. My general recommendation is that you should not attend a program unless there are at least 3 research groups you'd be very happy joining. 
On the other hand, you still haven't given any real information for any of us here to go on. Giving professors and a generalized description of what you want to do doesn't allow any of us to help you figure out how you would fit into those groups. 
Do the projects you want to pursue align with the research in those labs? Did you get along with other graduate students/the PI when you visited/skyped? Have you emailed and talked to graduate students in any of the labs?
None of us can tell you how well your background and experiences help you *fit* with any research group unless you tell us something about it.
All I can note is that while all of the groups you mention are technically "materials" groups, they're very different in the subset of materials research they do, the applications of that research, and the approaches and methodology they apply. It means it should be relatively easy to pick one over the other based on your personal interests, as they're quite different researchers.
You also didn't mention, but have you talked to each of those professors? Do they have room in their research groups, and are they interested in taking you on? Have they discussed what project of theirs you'll likely be working on?

There are a lot of academics out there who don't have the best social skills. There are plenty professors who can diplomatically tell a prospective student that they don't have funding/space* and think you should join another research group. But to me this sounds like a professor who doesn't really want you to join their group but isn't managing to give you an outright rejection. He's stalling for time ("go away and think about this"), bringing up multiple concerns (funding, hobbies, do you know what you're getting in to). I mean, accepting someone into a research group isn't difficult ("You want to join our lab? Great! Let's meet next week to talk about projects."). 
It's like when someone is trying to dump you: "I don't know if I want a serious relationship," is a face-saving way of saying "I don't want a serious relationship with you." Except in this case "Are you sure you want to do a PhD in my lab?" could well be a way of saying "I'm not sure that I want you to do a PhD in my lab."
 
 
* Sometimes academics use a concern about funding as a way of rejecting someone they feel would be a bad fit personality for their group. Because you can't argue with a lack of funding in the same way you could argue about "motivation" and "fit", and it avoids hurting the grad student. 

In the sciences, your dissertation research is often turned into publications as you go along. They're kinda two sides to the same coin. 
It is really lab & field specific. I know of science students who can get out in 4.5 years with a good list of publications, and who never have issues getting a job. But you also have to have a PI who supports you getting out quickly. Some PIs view 5 years as a fixed term, so even if you obtain good results they may still intend for you to stick around for the whole 5 years. 
It also depends a little bit on luck. You want to be assigned to a project that can generate enough results in under 5 years. If you are doing something that has never been done before then who knows if it will work or not? Sure, you could be in the lab 90 hours per week for 4 years, but be so unlucky on projects that you still don't get enough data for a thesis. 
As you start your PhD, my advice would be to talk to your PI about their expectations for successful completion (do you need x number of papers, for instance). Choose a lab where the students defend within a sensible time limit (you do not want to be where the 7th year PhDs are, under any circumstances). Work hard and work smart, but I wouldn't obsess about a 4 vs 5 year deadline: grad school is stressful enough without adding more arbitrary hurdles.

Personally, I'd deal with this by not dealing with it...at least right now. 
Your friend is upset. You are upset and rather stressed. Even if you apologise to her, you still have several more months where writing is your priority and will be taking up most of your time. 
I'd send her a short message saying (i) you are sorry that she's unhappy (ii) you appreciate her as a friend and know that you've not been able to give her your full attention (iii) right now you need to focus on thesis writing until April and it is detracting from ALL your friendships, not just hers (iv) how about we do something nice/fun together in April once you have finished with the crazy thesis stuff? 
If she isn't conciliatory after your response...don't waste more time and energy on it. Focus on quality time with your supportive friends (if any of them can try and gently explain to BF about grad school stresses & time commitments it might be better than you getting sucked in). It might be that your friendship has run its course, or you have grown apart with your different life experiences...but you can deal with that after the thesis.  

If you dress like a lazy undergrad in grad school...you'll feel like a lazy undergrad in grad school. 
 
If you dress sharp, smart and professional...you will feel sharp, smart and professional. And other people will treat you as if you are sharp, smart and professional.  
 
Burn the Uggs.

For crissake dude it's one random elective. Just fucking drop it already.
That said, your attitude towards failure has got to go. You're obviously not doing well in this class, yet you can't even admit that to yourself. You know 70% isn't a good grade. You know that you failed a 25% assignment, and you know that getting the precise percentage on your paper isn't going to make a difference to your outcome because that F is already pulling you down significantly. Instead of doing something proactive to fix your situation, such as dropping the class or asking for extra credit or meeting with the professor and having a frank discussion without hints and insinuations and fortune-telling on tarot cards, you're focusing all your energies on denying that this is happening to you. It's okay to fail. It's okay to feel challenged. It's good that both of these things are happening to you because it means you're working to your potential. One of the big goals of graduate school is getting you to stop needing validation, whether for practical or emotional reasons, from grades and professors, and instead teaching you to make decisions based on your own thoughts and evaluation of the situation.

I can see myself doing 1 of 2 options depending how risk adverse I felt:
(1). (Most risky) I would hold out for Lab 4 and then if the professor didn't get funding - do some new lab rotations in the fall.  My only concern with this is what happens with your funding for your second year if you haven't joined a lab yet?
(2) (Least risky).  Join Lab 2 - that seems like the least worse lab for you.  I know you said you felt like you were in prison (really - not good!) - but is it possible that the PI is just like that with new people to the lab and would back off to about a 5 on the independence scale?  Could you ask advance students in the lab?  If that is not the case - could you try to reset your perspective about this so while you have a more hands on PI that is not your preference, perhaps there are benefits to that that you don't see quite yet - i.e. less floundering/going down rabbit holes? 
 

A second childhood? A socialist utopia? Everyone else's graduate school experiences must be really different from mine.
Graduate school (and academia) is the real world. It's not some respite from it or an imaginary place; it's just a subset of the "real world" that functions in a space of privilege in many senses. It's no less or more real than being a hedge fund manager or a CEO or a corporate lawyer or a marketing manager. Those people also experience extraordinary privilege by virtue of the money and status that they have, but they still have "real" experiences. Reality comes in many different forms, and it's not all cold basements and dirty sidewalks.
I didn't feel like I was sheltered from "the real world" in graduate school; I existed in it. I had to pay my rent and feed myself like everyone else; I saved and worried about emergencies; I took public transit; I dealt with grumpy administrators and not getting paid on time and issues of being the only woman of color in my department. There are certainly privileges to working in academia, but there are downsides, too, and it doesn't remove you from living in reality. I think you should just enjoy whatever privileges you have, while remembering that there are downsides to everything and that you don't live in some made-up fantasy land but actual real life.
As far as academia being a socialist utopia...no, it's totally not. First of all, many universities get a lot of their funding from private interests, charitable donations, and contracts with corporations. A lot of the medical research going on on your campuses, for example, may be funded by drug companies (or overseen by drug companies, or bought out by drug companies). Many private foundations give grants to researchers to investigate issues of interest to them. And especially as you ascend up the research ladder, gaining the money and resources to conduct your research looks more and more capitalist. And it's definitely not a utopia - academics are no less likely to be grumpy and disagreeable than any other employees out there. They're just people. People who generally have a job they like and are passionate about, which is great, but people nonetheless.