It isn’t easy to get into a top 10 program from any school. If you do well in these programs, it is certainly possible. However that will be up to your performance. Just remember the top students from a lot of colleges will be competing for those spots.
Were you able to attend PFW at Caltech? That provides a lot of information about whether it is a good fit for you. I don’t know details about Berkeley. Here’s a bit of info on Caltech.
Research should be pretty easy to come by at Caltech. You have to put in the legwork to contact professors and write a proposal, but it is available, even for the summer after freshman year. About 1/3 of Caltech students participate in SURF each summer. It's perhaps a bit harder to get one in math than for other majors unless you are willing to go the computational math route. See http://announcements.surf.caltech.edu/index.cfm?event=ShowAOPublicList&inFrame=&type=SURF&formType=AO_CIT and search the page for math. Those are the "canned" research topics for summers. You can of course talk to professors about doing research with them. I can PM you an additional list for academic year research if you are interested.
There were indications that the median graduating GPA at Caltech was about 3.5 back a few years ago when they did away with "graduating with honors." That doesn't mean classes are easy. As I'm sure you know, they are quite rigorous.
I think currently about 1/3 of Caltech graduates go straight to a PhD program. That figure is lower than previously because of the popularity of the CS major and the amount one can make in industry. Anecdotally, a fair number of Caltech grads go to "top" grad schools.
If you are doing good undergrad work for a professor, I don't see why they wouldn't be happy to write you a great recommendation.
I don't know the general ed requirements at Berkeley, but Caltech's are substantial, so you should choose Caltech only if you are OK with doing a lot of physics, for example. Also, your AP and college courses won't count at Caltech, while they will likely count at Berkeley. (For Caltech, you'd take placement and/or place-out tests this summer for math, physics, CS, chemistry, and English.) The Core requirements at Caltech outside math would be:
3 quarters of freshman physics
2-3 quarters of sophomore physics (required as part of the math option): quantum physics and statistical physics
2 quarters of chemistry plus a chemistry lab
1 quarter of biology
1 menu class (one of intro geology, astrophysics, energy, or a few other options)
1 other science lab
science writing
12 humanities and social science classes (so 1 per quarter)
3 PE classes