Austin's and Matt's comments above are spot-on. I will just expand a bit on them here:
Regarding 1), as indicated in this thread, at the vast majority of US schools it would actually be impossible for a student to take 20 math classes by their senior year, as there are so-called "distribution requirements" that must be satisfied. Even the most dedicated, top US math students will have probably have taken less than 20, so don't feel pressured to attempt more because you feel you have to catch up. I should caution: make sure you are learning the material well! It is far better to take fewer math classes and really absorb what you're learning, than take a whole bunch, if the latter means that you are going to miss out on key aspects of the math. Good grades do not imply full comprehension (and vice versa, unfortunately).
If you still want to do more math anyway, I recommend simply reading on your time! It is a fun and free way to teach yourself some math that perhaps your university doesn't offer a class on, or review something you didn't get fully the first time around. You might also be able to audit a class you think would be interesting, but that you don't think you'd have time to follow fully.
Regarding 3), the frontier of mathematics is so vast (and always growing) that there is never any danger of the best people somehow "taking" all of the research. Perhaps you might not win the Fields Medal, but you will be able to make many original contributions to mathematics. It's best to focus on the enjoyment you personally derive from doing math; don't worry about the fact that there are X number of people "better" (whatever that means) than you. And you should take solace in the fact that mathematics is an extremely collaborative field - people are more than willing to share their knowledge with the community, as this site evidences quite frequently. There is no denigration of people for knowing less; we are all in it together.
Also, take a look at Terry Tao's blog post on this very issue. (Actually all of his blog posts are very positive and informative, I recommend reading them.)
There are also relevant posts on this site; the main one I can find at the moment is here.
Regarding 2), I'd like to know the answer too, as I'm also currently applying to grad schools :) But given your description, my (unqualified) opinion is that you have basically done everything for a grad school application as well as you could have done (your English looks excellent, and I assume you have gotten superb letters from your professors!) There may be some difficulties at US universities due to your status as an international student; I'm just guessing, but some places might have a quota. But even if that is the case, there's nothing to be done about it.
Again, this is my cursory, uninformed judgment; there are many professors here who have served on admissions committees at their schools, and you should not hesitate to listen to them over me.
As this is a community wiki post, I will shamelessly include Gerry's advice above, that you apply to plenty of so-called "safety schools" (i.e. schools that, while perhaps not as high a caliber you would like, you feel quite confident you'd get into).
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To me, grades aren't the whole story. Some profs have individualistic grading ideas and may be particularly hard on the one thing you are worst at -- even if it isn't such an important thing. Or maybe the course was a stretch for you and an A- shows significant progress. Who knows?
What is much more important is to learn the material. Now by "learn" I don't mean get an A. I mean having a deep understanding of the subject. You could test this for yourself in a variety of ways. First, given a problem similar to one you've seen, can you do it? Can you do it easily? Next, given a problem in the subject that you haven't seen, but appears to be in the topic area, how much progress can you make on it? I wouldn't say, can you solve all of them -- your first try could well be one of those that look simple and are terrifically difficult. But can you get anywhere? And can you solve a good percentage of them?
How many questions do you have about the course material? If you think you understand absolutely everything, then you probably haven't learned enough. If you've got areas that seem fuzzy, or questions you hope no one ever asks you; or if you have dreams that you are failing a similar course -- you have questions. Drag them out into the open and get some light on them.
In general if you want to do very, very well at math, you need to work a lot of problems. Theory is all very nice, but there is nothing like trying to work a problem to demonstrate that you really didn't understand it. The more problems you work, the more you will know. If you get stuck, you can ask -- one of your profs, or post it here. There are bunches of eager, knowledgeable people here who enjoy helping out.
Re graduate school, quite a few will take you with less than a perfect average, particularly if you demonstrate a lot of knowledge. Some schools have a "come if you wish, stay if you can" philosophy, and basically let anyone in (no matter what the catalog says). Some of the anyone's are certainly not able to stay, but the philosophy is that if someone wants to learn they should be given a chance.
Re number theory vs diff eq, there are those who are impressed with one and those who are impressed with the other. I personally prefer the applied areas, but that is not a majority opinion. As to whether one is more helpful than the other in getting an REU, I suspect it would depend on who was reviewing your application and what his/her prejudices are. Take both, eventually.
REU's are very competitive, and I don't know how many go to freshman. You have 2 choices: you can apply and maybe you will get one, maybe not. Or you can decide not to apply and guarantee you won't get one. I say if you have nothing to lose, go for it.