The GRE is primarily calculus, a near-perfect score is somewhat expected. The lack of research will hurt you at the upper level, as well as your classes if they are not at the graduate level. If you're a woman or minority that will help a lot.
You'll probably be accepted at UVA. I went there for my undergrad and with a slightly higher GPA, research, and several graduate classes, I was accepted easily (it was my safety school). The incoming class that I met at orientation had stats similar to yours or slightly lower.
My undergrad advisor said that I would be able to get into places like Berkeley or Michigan, but my application process was a bloodbath and I ended up at University of Maryland (which turned out to be great). I hate to be a killjoy, but it seems that things are a lot more competitive since the days of our professors. UIUC may be a stretch I think, I have an officemate who came from there and didn't get in. My advice is to shotgun the entire band of universities.
Don't be caught up in the ranking though, it's much more important that the school has strengths in the subject you want to study. You might want to go to a place with a decent applied department, because you may end up doing it. I came to UMD as an algebraist, and I was incredibly lucky that there was a good applied program because I ended up doing that.