rlz567

If you have your own self-named grant or your senate father is willing to give you a strong recommendation 

The GRE is more like a filter. A high GRE Q won't get you in but a low GRE Q will disqualify you. With that said, I strongly recommend you spend some time studying and retake the GRE. It should be pretty easy for you to get 168+ since they are mainly high school math. You can also consider retaking the GRE math subject test. I am saying this because with your math background and some preparation, a 90% is very attainable for you.

Well, let's start with this:  if you could get your quantitative GRE score up to 164, I think you'd have a decent chance for most schools except the very top (Stanford, maybe Cal, Harvard, and UChicago?).  I might be overstating your chances at each place, but I think you'd have a decent chance to get accepted at UWashington or a school of that level.  I can't really judge your research, but you might be in even better shape if it's particularly good work.  As a note, you don't really have to submit a Math GRE score except for Stanford, so unless you're particularly interested in going there, you probably don't need to worry about that.
With your current GRE score, it is going to hurt a little bit, since sometimes having those high math grades but low quantitative GRE scores makes people wonder about the rigor of the classes/grading scale thereof.  At a top-50 school, though, that won't be a big concern -- even less so if your school has a strong math department.  As such, it might hurt you a little bit by comparison to other applications, but I feel like some of those schools would still be inclined to "take a chance" (if we can call it that) on someone with your grades.  I think that UWashington to Penn State area, maybe down to UCLA or so in the statistics rankings, is a good place to concentrate applications.
Overall, I can't really recommend schools for your research interests (although UNC statistics seems a likely candidate), but I think you can apply to more of the elite programs you're interested in.  I'm not the expert on profile evaluations or this kind of math, though, so listen preferentially to the more senior posters if they drop by.

Brown doesn't have a statistics PhD program, so I'm not sure which program you're referring to (applied math probably? Biostat?)
I definitely would not submit your subject GRE score.
I'm going to be a lot less optimistic than the above.  I think your 161 GRE Q is going to really hold you back. Yes, you have good grades, but grad-level course grades are known to be heavily inflated.  As said above, this would be mitigated if you were taking grad classes at Harvard, but not so much if you're at Villanova (49).
I could maybe see a big school like TAMU taking a chance on you, but I think you're going to struggle getting into a top 20 program with a 161.  I would do whatever possible to raise that score - even a 163 would help you significantly. I think Washington, Cornell, Brown and NYU are not realistic targets.  If you improve your score to a 165, I think they could be in your "reach" list and schools like Florida and PSU could be targets.