The situation you described is quite problematic and (having heard only one side of the story) seems to involve misconduct by your advisor - as well as lack of perspective and/or complacency on your part as you've been faced with her behavior.
But let's start with what you explicitly asked and move on to more general suggestions/advice.
I want to know how much authority does a PhD adviser have over your thesis.
Ethically, little. Your advisor should only verify that what you intend to submit is academically acceptable as a Ph.D. thesis: Sufficient contributions, adequate novelty, work which you did either yourself or as the primary involved researcher, no plagiarism, errors etc.
Technically, however, in some universities you require your advisor's explicit authorization to submit your thesis. You need to check whether this is the case in yours.
Even if that requirement is in place, however, it's not as much as a hurdle as many people worry it might be. if you have a PhD thesis written up, that you've gotten a bit of review on to correct any glaring problems, and that you've typeset and maybe even printed a copy you can wave at people - it would be quite problematic for an advisor to say "no, you can't submit" without a valid reason. If you contact your department/faculty's Professor in charge of graduate studies and wave your thesis at him, saying "My advisor won't let me submit", and with this being after 7 years - significant pressure is going to be put on her.
Can she ask me to remove important sections of my analysis that has been with her for over an year?
You're putting emphasis on an invalid point. Nothing has "been with her", it's in the draft of your thesis. It can theoretically be her opinion that you should remove something. In the case you describe, obviously this seems unjustified. The main issue is the justification.
Also, you are not at all required to oblige her and you can put whatever you like in your thesis (under the conditions I mentioned above).
I am asking this because of a strong suspicion that she might want to scoop my work once I leave school. In that case the only proof of authorship of my work is the thesis document.
Then publish that part of your work somehow before she has a chance to plagiarize it. As publication does not preclude work from being part of your thesis, it won't hurt you in that respect. Also, if proper, peer-reviewed publication of this part of your work time-consuming/difficult/prolonged/won't work out for some reason, go for a pre-publication - on a website like ArXiv, or on your personal website - as some sort of a technical report. That should be enough for her not to be able to plagiarize your work.
If I have to press forward with my work, how should I diplomatically go about it?
Make sure that your friends and colleagues know about the situation you're in, sans your suspicions of your advisor. Specifically, make sure whomever of them who can understand your research knows exactly what you've done yourself (so as to be able to back up future claims of plagiarism). That last bit may also be pertinent to other senior academic staff members, so perhaps find the opportunity to bring it up, e.g. when sniffing potential future collaboration. Maybe schedule some colloquium talk about it.
Also, contact whoever is in charge of graduate studies in your department, and tell them you are having trouble with your advisor - without making accusations for now. Talk about your efforts to finish up and that it doesn't seem like your advisor is helping you get there; perhaps also talk about things that happened over the past several years before the thesis business came up. Be sure to mention at some point that part of your work that you're worried might be plagiarized - of course not this worry of yours, just try and make sure it's clear that you've had results doing whatever it is you did.
No less importantly: Talk to your graduate researcher's union. When I was vice-chairperson of our (half-recognized) union, a PhD candidate came to me with a similar situation: I promised him what your union should promise you, which is that if actual foul play begins on her part (like publishing your work under her name), we would go have a talk with her on your behalf. And if she refused to rectify things, there would be a massive scandal; her name and picture would be up on posters all over campus branding her a plagiarizer; we'd get the local media to report on it; we would file disciplinary charges (not that management would entertain them, but who cares); we organize protests in her classes - whatever it takes.
... but all this depends on actually having a functioning union. If you don't, try the same with whatever graduate student organization you do have, although then it's less likely you'd get the kind of backing you need.
As for your interaction with her, I don't think you've given us enough details for me to be able to make suggestions about that.
Finally, I think you've mishandled the situation so far. You see, your Ph.D. is your own project. Your advisor merely advises; you carry out research, you write it up, you publish, you submit. A Ph.D. is an exercise in being an (sort of) independent researcher. And feedback is something you can, and should, get from other people - in your department, your university, and elsewhere. You should go to conferences, especially if you're in the US where they're close, even when you haven't published there sometimes. You should have tried, at least, to form relationships with people in your field that are not mediated by your advisor. If you had done that, her lack of attention (or even abuse) would not have had that much of an effect.
Now, true, if we're talking about an experimental lab setting, then the head of the lab is more dominant in determining whether your research life sucks or not; and I know it's easier to give this advice than to act on it (and I'm struggling with the same challenge myself even now, as a post-doc) - but it sounds like you've been mostly depending on her support and feedback. I'm sorry you're having to change that under such unfortunate circumstances.
