Strong points:
General GRE- 800 Q, 700 V.
GPA of 4.0, with almost 60 credits in math by time of graduation, including 2 semesters of analysis, abstract algebra, linear algebra, differential equations, graph theory, and one semester each of number theory, topology.
Will have very strong letters of recommendation. My letter writers have suggested I apply to Princeton and MIT. I have done independent studies with my writers so they know me quite well.

Weak points:
Subject GRE- Don't know yet, but likely between 650-780 (65%-85%)
My university is ranked near 105 on the NRC list. It's fairly unknown.

Based on that information, would my application be considered relatively strong? What types of schools should someone with my level of competitiveness apply to?

So far, I've decided I will probably apply to these schools:

U of Michigan
Rice
Berkeley
Rutgers
U of Washington
UT Austin
Northwestern
Princeton
Maryland College Park

I'm having difficulty choosing which of these to eliminate from my list. I think I should add a few safer options, but I'm having trouble finding a lot of information. I'm using the NRC and US News & World report rankings, and then investigating department web sites, emailing admissions directors, and so forth. I'm about to try to email a few grad students at some places. But perhaps some people here could give me a few recommendations.

I'm curious about the atmosphere at each place, the relationship between students and professors, how friendly it is, and things of that nature which cannot be found on department websites. I'm somewhat ambitious, but prestige is less important to me than learning atmosphere. I'm afraid that an unfriendly atmosphere may hurt my demeanor. I'm highly motivated and can study by myself and go much faster than classes usually do.

I don't know what I want to specialize in, but I have found the following subjects attractive: algebra, algebraic geometry, algebraic topology, category theory, knot theory, topological field theory, lie theory, geometric group theory. That list is not exhaustive; I also enjoy analysis, dynamical systems, ergodic theory, number theory, lots of applied math, and many other things. But the first list seems to attract me most (at least for now).