Okay, first off, I think this might vary a lot between pursuing a Masters and PhD, I went straight to PhD so I can only speak from that experience. One thing that I think goes for both is that you need to have established some kind of relationship between yourself and the adviser you are seeking to work with. Read their work, know their research, and you should like them/feel comfortable with them as a person too. You'll spend a lot of time communicating with this person so you want to be as sure as you can be that they will advise you in a way that's productive. Like, some advisers are really hands off (like mine) which is good for some students but terrible for others. You want someone who will read and comment on your grant applications and respond to your emails (this seems obvious, but I assure you it's not).
So yea, for me, I only applied to programs that I emailed with and met with the person I was proposing to work with. I didn't want to waste time composing shoddy applications and I tailored each app to the person I was writing to. This means I literally changed details of my project to be geared to those people, and sometimes cheesily citing those peoples work where it fit---this makes no difference in the long run because your project will become something very different from what you are writing in your statement. I don't think advisers accept students who they solely read their app but haven't communicated with. Do all that cold emailing, it works in academia, everyone is used to it.
Other than that, everything you are doing is good. Money matters, but it's not the only thing that matters. I took the offer that was giving me slightly more money and the institution I believed I would have access to more grant money, but now, sometimes I think I could be happier at my second choice (so it goes...). Some programs make you TA, sometimes you can get NSF or other grants so that you can pass over institutional funding and not TA. Personally, I love teaching, it's the only concrete aspect of success that I've encountered while pursuing a PhD thus far and it breaks up the monotony of research and writing.
GRE is highly dependent on the program/world you're in. I took a crash course because I was worried I'd fuck it up and when I met with potential advisers they were like "we don't even look at those". In my field, they care about one's writing (sample + statement), past academic success, and recommendations. You always need to hound your recommenders (something you will get very used to and tired of). Once they submit, you can make them cookies or make them a card. I don't think gifts are appropriate but if they put time into the rec, it's nice to acknowledge their effort.
Also, definitely feel free to send me your statement/resume. And on that, send it to anyone who you like their writing and is familiar with your stuff because it's very useful to have a variety of insights. I do this regularly now, but I basically wrote my apps the day before I submitted...