Here are my thoughts:
1. You'll probably get the best advice from your adviser or a professor at your institution. Maybe you could take a grad course in probability after you've completed grad real analysis or look into some stats grad courses. Personally I'd just take whatever seems interesting.
2. Any decent program should provide you with full funding (i.e. tuition waivers and a stipend via a TA position or fellowship) regardless of your state residency. You should view an acceptance with no funding as a "soft" rejection.
3. I recommend taking the subject GRE the April before you apply and see how you do. If you're satisfied with your score then you can work on other aspects of your application, and if not then you can sit for it again in September or October later that year. You'll want plenty of time to review and practice solving problems quickly, so I would start studying two to four months before each sitting. It's my understanding that the general GRE isn't as important for applications, but you still want a decent score. It depends on how confident you are, but maybe start studying a month or two in advance if you want to be thorough.
4. An REU should make your application more competitive and may also give you something interesting to talk about in your statement of purpose. I imagine coming from an underrepresented group would give your application at least a slight edge. The MAA maintains a list of resources
, including summer programs, which you might find helpful.