There are no general rules for resumes, it all depends on what else you have. In general you want to keep it short. If you've done a lot of stuff recently, leave off the old things that aren't as impressive - they will only detract from the overall impression your resume gives. The older something is, the less anyone cares, and the less it should be represented. The exception to this is things that are more impressive should last longer than things that are less impressive. If you scored okay but not great on your SATs, I would take them off once you're in college. If you scored perfectly on your SATs, I would leave that score on there even after graduation from college. The same goes for something like being a high school valedictorian - keep that on there for a while, but drop everything else about high school.
How far back you need to go and how much you need to stretch depends on what you've accomplished recently, but don't stretch back in time to fill more than a page. One page is typically the ideal length for a resume, although for technical positions it's often okay if you go over, and for academic positions they'll want a longer CV.
There is no right answer to your questions since it depends on the overall impact your resume has and the impression it gives. I would recommend speaking with your career services office and also having everyone and their cousin review your resume. If you can synthesize all of the competing opinions you get and pick out the best ideas, you'll be able to craft something that is the most likely to appeal to the one or two people who read and subjectively judge your resume in the end.
It's not a bad idea to spend some time on this now so it's a lot easier for you when it comes time to find a real job. Having a resume also gives you some perspective on what gaps you need to fill in and what you want to accomplish over the next few years to get to the job you want.
You ask some good questions about what to put for a math specific program... and I don't have the answers

. You'll have to make a judgment call. It may be easiest to put everything on there, and then, in a new file, with the help of some other people, pare it down to what has the best impact. It's good that you're keeping your audience in mind though, and you should typically never use the exact same resume for different positions. You can always tweak for your current audience. Focusing on math is definitely a good idea in this case.
You can search for the career services websites of various colleges for some good advice and sample resumes too. See http://www.trincoll.edu/depts/career/resumeguide2006.pdf , for example.