Crystalline_Sunlight

Yes, please please talk to your advisor about this. Usually the majority of reasons why a student is removed from a program is an issue that could have had a chance to be solved by communication early on. I know this is easier said than done because most students fall into the trap of thinking "I don't want to bother my advisor unless there is a big crisis (i.e. actually about to be kicked out)" and then when a crisis happens, it's too late and/or the student doesn't know the advisor well enough to open up about an issue.
Good luck!

My question is this: suppose he tells you you have to figure it out on your own. That's possible. You're already in this position of figuring it out by yourself, so what have you got to lose? So of course this is one possible outcome, but even if this is how it ends, at least now he knows so any future conversation can build on it and your efforts to improve. I think that coming from a place where you say "I've struggled in the past and that's affected my performance; I'm working to improve my situation but I worry that my past has made it slower and more difficult, and I worry about the possible implications of that. I love my program and enjoy being here, and I want to make sure that I can stay" is something that he can at least understand, if not relate to. 
And of course there are other possible outcomes to your conversation, like that he can think of ways of helping you out. Maybe he knows more experienced students who've gone through this and can give you feedback or support along the way. Maybe he knows the professor in question and can intervene if there is a need. Maybe he can help you draft an email if you ever need to write anything formal that it'd be good if someone looked at before you send it. He's your advisor, hiding things from him isn't a good way to start the relationship. 

It seems you're interpreting your struggles as an issue between you and your program instead of understanding your own individual struggle as a student. I don't think there's a point in emailing your professor that you realized what went wrong because finding out your mistakes and learning from them is a student's job. I would show your professor that you're capable of doing well in the class by doing better on future assignments and your final. It's also more of a student's job to be proactive than the adviser's. I don't think anyone should expect advisers to contact students and make small talk about how you're doing in the program. Professors are busy people and can have odd schedules. If you need to speak to your adviser about something, contact him. If he's away, thankfully, there are many other forms of communication other than meeting in person.
Everyone goes through a learning process in finding out that grad school is different from undergrad and that it requires students to make adjustments. Although happiness is important, you should start thinking about grad school in a more professional manner. Think about your career goals, how attending your grad program will help you achieve them, and how to get the most out of your grad program, not about what makes you happy, sad, hateful, etc.
 

1. I don't think you should email the professor, but you could certainly come to her office hours in a week or two to talk about your learning process and whether these new insights you've discovered are on the right track, as well as if you're following the concepts in class currently. 
2. I would recommend talking to your advisor. Right now the problem is small and hopefully fixable. The more you wait, the larger it could potentially become, and the harder it will be to ask for help. You should let your advisor know early on so he can help you in any way that he can, which is his job. I don't think you want to be in a situation where at the end of the semester he suddenly discovers that you've been struggling for months without letting him know, possibly with consequences for your ability to stay in the program. Talk to him now, even though it's an uncomfortable conversation, and get all the help that you can as soon as you can, so you can fix this. Knowing who and when to ask for help is a very important skill. 

I had gotten to class early, no one else was in the room, I looked at my phone to check my email, saw my acceptance letter and began jumping up and down like a crazy person, doing the happy dance. Though I was alone in the room, the door had a window and a few people walking by definitely caught me in the act Luckily I managed to regain my composure before any of my classmates walked in 

You can avoid mentioning that you think her research is lacking the breadth you want. Just say you are particularly interested in projects that connect to the bigger picture, such as <example>, and wonder if such a project would be available there. She will be able to read between the lines.

You could certainly frame it in the context of asking for a collaborative project, but I think it might come across better if you propose a project that you would be interested in undertaking, and asking if that would be a possibility under her mentorship. This way, you would come in with an understanding that there are opportunities to do the kind of research you are interested in, and she would not be surprised concerning your focus. You say that you're interested in connecting to the bigger pictures implications, so propose something that would demonstrate that! Congrats on the acceptance, too

Thanks for the advice all Looks like I'll be visiting after all.
 
In other news!! Just received an invite out to Cornell's Visitation Days for Prospective Students :DDD It's definitely not an official acceptance, I don't think it's even an unofficial acceptance. Anyone know what this means for my chances? Surely they don't invite all prospective students...
Will this mean that depending on how the visit goes, it could influence whether I get accepted or not? Cause that puts a lot of pressure on it! 

I was accepted to UCLA! Yeaahhh!! Received an unofficial email from POI a couple days ago and then the official acceptance email the next day. Fully funded! 
 
I have a dilemma though. Their open house is on February 17th meaning I'd have to miss 3 days of class. It wouldn't be the end of the world, but I'd be missing some important stuff. Is it worth it to go? I've already visited the campus last summer when I first met with my POI... Also I live in a town whose airport literally has one gate so I'd have to drive 5 hours to the next big city to get a reasonable flight...Worth it? They want an answer today :-/ 

Same! Though it is a relief that there's officially nothing more I can do--no one else to contact, no more researching schools, no more editing my statement. I've given it all I've got! Now it's up to them...