Eigen

As general, non-field specific advice, I usually tell people this:
If you are perfectly, or reasonably over, qualified for the jobs you're applying for AND putting at least 4 hours a day into thorough applications expect about a 3-6 month turn around for most companies, 4-8 local agencies, 1 year + for government.
At the highest point of my job searches, I think I average about 20/week and regularly follow-up with the 5-10 I actually care a lot about.
 
Every once in a while I hear about someone landing a job much earlier.. Usually through networking. 

From your posts, it seems that you are overlooking a lot of important factors when it comes to graduate school and individual programs. The structure of the program ACTUALLY matters, so i would look into those ASAP. Also, you can't actually be neutral when writing your SOPs. There must be something about the school that you care about that is not simply the fact that there's professors that do the work you want to do. If you say that during the interview, they'll know you don't actually care about the school and you'll likely be rejected. Across all my interviews, the most common question for me was "Why do you want to come here?". It is also worthy to point out that, though it shouldn't be a main factor in your decision making, name brand matters, especially in academia. Top schools tend to not only have more resources, but they also have a wider network of connections that'll most likely help you land a job--- so take that into consideration. 
PS:  Scripps level of prestige is comparable to Harvard's, and the competition there will also be intense. Thus, if you question your ability to get into Harvard, that applies to Scripps as well. 

Trying to do anything for this fellowship (or honestly, any grant) without sounding cheesy is.... really difficult. Embrace the cheese!

To me, if I got it cold from a student, that second paragraph reads very abrasively. It comes across less as interest, and more as questioning the validity of the work and asking them to justify what they're doing and why. 
Especially leading into your third paragraph, where you basically assume they'll respond. 
Nowhere in your message do you mention anything like "I'm sure you're busy", or "if you have time, I was curious"- things that take the tone from entitled to polite. 
But now we're also getting back to the "why are you doing this" question. What are you hoping to gain from it?
Introducing yourself through email doesn't help much with admissions in chemistry, and if you set the wrong tone it has the potential to hurt. You don't want to get lumped in with the large number of people that send what are basically form letters to a bunch of different faculty that everyone treats as spam. 

Just to add, I think the difference here hinges on not your training from your advisor, but what the travel grant required. 
If the travel funds were contingent on you collecting data, then I think it's within rights to require you to either get the data or reimburse the travel funds (basically what they're asking you to do). 
I know most granting agencies are pretty strict about data management.
Also, to be clear, since @TakeruK and I seem to be interpreting it differently- is it your advisor that's requiring this, or the director of the site where you worked this summer?

My point was that you don't need to know if they're accepting students before you apply. That's the whole point in picking schools with multiple potential PIs. 
I can't see what you gain by asking them that now, and think you'd be better off introducing yourself, saying you're applying, and asking an insightful question about their work. 

Honestly, only some of this needs to be done before you apply. 
You don't need to know all of the faculty are taking students/have funding when you apply, just before you accept an offer. 
And if you take out those parts, it's easier to contact faculty without "over-selling" you wanting to join just their lab. 
Then you can send a more general email as a student who will be applying, and is reading out to faculty who they have particular interest in. 
My personal advice is... Don't contact someone unless you have specific and relevant questions to ask them.
Ask about possible directions for their work- something like "I'm really interested in the work you did in X paper 2 years ago, and was wondering if you have plans to pursue Y direction?"
You can include some things about yourself and your background as you ask these questions, which comes across less pushy than an overt "sell". 
If you can get a conversation started, you have more opportunities to sell yourself down the road. 
Keep the first email short and make it have a specific point. 

To me, if I got it cold from a student, that second paragraph reads very abrasively. It comes across less as interest, and more as questioning the validity of the work and asking them to justify what they're doing and why. 
Especially leading into your third paragraph, where you basically assume they'll respond. 
Nowhere in your message do you mention anything like "I'm sure you're busy", or "if you have time, I was curious"- things that take the tone from entitled to polite. 
But now we're also getting back to the "why are you doing this" question. What are you hoping to gain from it?
Introducing yourself through email doesn't help much with admissions in chemistry, and if you set the wrong tone it has the potential to hurt. You don't want to get lumped in with the large number of people that send what are basically form letters to a bunch of different faculty that everyone treats as spam. 

But that's not what you said- you said you can't establish residency while in school. You can establish residency while in school, you just can't immediately become a resident for tuition purposes. If you do establish residency while in school, your tuition will go back to the in-state value after you've been in the state for a year. 
Most undergrads don't establish residency because they can't meet the requirements- and it's usually not worth it. It means they (usually) can't be on their parents insurance, can't go back home and work a job over breaks, etc.
Most graduate students do establish residency, but don't have to- they can opt to consider themselves students and go home to register a car, get at drivers license, etc.

Wow, I am sorry for opening up the "what makes a professor" conversation. OP, just take away that the people who taught you may not have specifically had a PhD (and that isn't a bad thing, just maybe focus on seeking advice from those who do have PhDs when it comes to questions about applying for PhDs).
Good luck on applying to grad school!

ALCON--
When picking a screen name at the Grad Cafe, please consider the value of NOT using the same name that you use on social media. For some members, especially newer ones, finding out who you are is as easy as highlight ===> right click ===> search google for <username> in less time than it takes to read this sentence.
While it may be highly unlikely that a member of an admissions committee will come here looking for a specific person, the Grad Cafe is a known location, and the contents of the fora turn up in Google searches. While you may think that your personal interest in X is no big deal, you will never know if that interest is a deal breaker for departments that have been burned by aficionados of X.
 

I got mine at the end of my first year, and deferred one year, and wish I'd deferred two. 
Using it later gives you more funding when you need it- use available funding your first year if you have it. 
First year is the best time to TA if you need to as well, at least in terms of research productivity. 

Of course each thesis is unique (otherwise it wouldn't be making a new contribution to the scholarship). But, there are commonalities in arguments which it would behoove you to identify long-term (Note: not entirely necessary for the writing sample of a PhD application). If you really want to get better at your argumentation, then it might be helpful to map out how each article/chapter you read works. That is, what's their thesis, what evidence do they support it with, and how do they use that evidence in making their argument? It's not just about language but also seeing what texts/sections people are drawing on when writing about your topic.  
So true about only a Professor being guaranteed to have tenure. There are Associate Profs at my institution that don't have tenure yet (often because the tenure and promotion processes are separated, especially when one comes in with credit of years spent at a previous institution). 
There are also Teaching Professor tracks at many schools now. In fact, I think the semantic move to "Teaching Professor" is precisely because some people devalue people who are "merely Lecturers" (and there's been evidence of that in this thread!). The University System of Georgia has a teaching-track where one can be promoted and receive tenure, just as there is a research track and the combo research/teaching/service which has historically been most prevalent. There are also tracks where Lecturers can be promoted to Senior Lecturer after six years (so the same timing as going up for tenure if one were on the tenure-track), suggesting another avenue of permanent employment that belies how people typically think of the term. 

Trying to do anything for this fellowship (or honestly, any grant) without sounding cheesy is.... really difficult. Embrace the cheese!

Pardon me if this overlaps/repeats what someone else has said (shamefully I didn't read all the comments, so sorry). But if you are having problems coming up with an "original idea" (and let's just sidestep the debate about whether it is important that your idea is new, I'll just say that if your idea adds nothing new it's not adding to the conversation), I would suggest thinking about research more in formulaic terms.
 
First: you do need to do at least some reading. No one should expect you to read up on everything, but you should be reading up on 
1. the major works in your field (sadly, the cannon)
2. the well-known works in your subtopic
3. the articles/critics/contributors that you either admire or completely disagree with (I often find these to be very specific articles from major journals in my field, imagine these to be the most recently published out of the 3 types listed here).
 
This seems like a lot, but really imagine it 3-5 books and 6-10 articles. You shouldn't have to read all of it, skim the articles, get the main ideas, and pull out the applicable quotes. Read the books and then maybe read some articles/commentary about the books. There's your lit review. 
What I like to do with the lit review is instead of just summarizing the texts (boring and not always useful), is to draw out a bit of a diagram (it can be a list, a map, or just some notes, whatever works for you), about how the texts relate to each other. After doing this I look for the physical gaps. The recent articles will be the best indication of what has/hasn't been written about these texts, since they were recently published. Most (good) journals aren't going to publish articles talking about outdated/irrelevant topics. Most (good) scholars also won't be revisiting these topics unless there is still more to be said. The gaps that you find among the texts are your research topics. Once you have your idea, I find it best to chat with a couple of people in the field (if you have some professors who are open listeners). If you're idea is not new, they will probably tell you. I find that when I talk to my chair about my ideas she often throws a few more ideas for readings/tosses me a book off her shelf for me to read on the topic as well.
 
Once you have your topic, have done some reading, and are ready to start writing I find that following a formula of sorts can be helpful as a jumping off point:
 
What is your topic? This is the subtopic you picked.
What has been said about the topic so far? This is the review you did (don't always feel obligated to do the summary/traditional lit review thing, feel free to just mention a few of the authors and give a very short mention of their contributions. I've seen published authors do their lit review in a mere 3 sentences and it's brilliantly simple to mimic).
What is missing? This is your gap that you just found.
Why is it important (the exigency). Why does the gap matter (this may be the hardest part to explain, but your topic must have some sort of relevance to the field- does it answer some questions? Is it applicable to understanding other texts? Does it somehow apply to the classroom or academy at large?)
Then dive into your conversation. 
 
I apologize also if this is all very obvious stuff, but for some of my graduate coursework this is how we were taught to get started on research and I found it very helpful. In general I mostly just find it very useful to read though recent articles in journals (which it seems like you do as well) I would want to be published in. What are they writing about? How are they writing, and what "big moves" are they making in the text?
 
Also if this is relevant, I'm in Rhet/Comp, but I framed this as English/Comp in general. 

Hah, no jab at VAP's intended- I'm one, after all. 

Hah, no jab at VAP's intended- I'm one, after all. 

Really? Are we really going to go the "appeal to authority" angle? 
If we are, well, I'm a prof and I understand what @EmmaJava is saying, even if I don't agree that calling oneself "professor" when one has not been appointed as such is an act of fraud. It's clearly not. I do think, though, that we need to be as honest as possible about how universities are functioning, and who is doing the teaching. When I did not have a PhD and was teaching a ton of undergrad courses as a TA, I was always very clear to explain to the students that I was not a professor. I felt it was in their best interest that they have as much information as possible about how the university saw fit to handle their instruction. The literal definition of "professor as someone who professes" was somewhat immaterial in that situation. (How does one define "profess," after all? Clearly not every teacher at a university is a professor, and neither is a PhD teaching at a middle school. Context matters.) 
(And somewhat unrelated, where I initially taught, the students refused to call PhDs "doctor." They felt that professors were not doctors and did not deserve the title. So "professor" was the best you could do.)

Watching someone argue with a  professor about who deserves to be called professor under what circumstances is a bit surreal. 

Watching someone argue with a  professor about who deserves to be called professor under what circumstances is a bit surreal. 

Watching someone argue with a  professor about who deserves to be called professor under what circumstances is a bit surreal. 

Watching someone argue with a  professor about who deserves to be called professor under what circumstances is a bit surreal. 

@EmmaJava, there have been lengthy discussions about this in the Chronicle fora and elsewhere. But, to be clear, the origin of "professor" is that of "one who professes" aka, one who teaches. A graduate student teaching a class is operating in that role. So is a lecturer teaching with their MA. So are those with terminal degrees in art, music, etc. (you know, those with a MFA who teach and are tenure-track or tenured faculty). There's certainly a political/labor argument to be made (see here, for example) for not calling adjuncts or graduate students "professor" but that really isn't the point you're making. So, my question to you is what title of respect could or should one ask undergraduates to use for a TA who may only be a year or two older than them (which is also a situation most traditional undergraduates have never experienced until college)? You probably also want to keep in mind the gender dynamics often at play, which lead to female instructors being called "Mrs" while more honorific titles are used for male instructors (see here).
P.S. It's also worth noting that what I've said above and what you wrote are US-specific. In the UK and Australia, one's first tenure-track job with a terminal degree is as Lecturer.

Hey there! 
I just started classes this week and I can definitely say I've reframed my whole way of thinking. The whole being older thing seems to matter not one bit and I'm taking my boyfriend and daughter to a departmental BBQ this weekend. My cohort is very nice and supportive and we are all commuter students, so it seems that social outings will be well planned but worthwhile. Everyone knows I'm a parent and even though I'm the only one among the group it seems like it's no big deal. I'm very happy with the group I have. Given that we're all commuters I'm actually considering holding some sort of social event at my apartment now to get us all together. Long story short I worried quite a bit more than I should have :-)

When I said that it was in response to the person saying they needed to establish residency for in-state tuition. Sorry for the confusion....to clarify, what I meant is that establishing residency for tuition purposes usually involves more than living in the state for 12 months when you've already enrolled in a program as a non-resident. I think some people mistakenly think that having a lease for 12 months in a new state will be enough but often it is not. Some schools require proof other than just your name on a utility bill or on a lease. They want to see you're working (not just going to school) in their state so they will ask for tax forms showing you have been working in the state. Or they ask for proof that you own (not lease) property in the state. Bottom line is that most schools are becoming more strict on who they consider a resident. These strict requirements are usually for the students that are trying to change their residency status with their school. Hopefully that person's school is more lenient with making that change. I just wanted to let her know that it might not be that easy.