To the OP: I wouldn't underestimate your own talent and interest in mathematics if I were you. As you say you take around 5 math courses a semester that probably is, at the least, as hard as your experience will be the first years of graduate school (depending on the graduate school). At MIT, you probably won't get very much sleep (as I believe you only have 4 years). I also wouldn't overestimate the quals. I have looked at quals for some universities (and I am undergrad) and after taking 27 hours of graduate credit, I can do many of the problems very easily without studying or even when the exam tests material I took a year and a half ago. Of course, it depends on the university. As long as you find a good fit for a graduate school the quals will not be all that terrible. Besides what the quals do is give the opportunity to learn three areas of mathematics really well. It gives you a strong working knowledge of math and this is essential because it demonstrates that you can learn more advanced topics in these areas but also shows that you can learn a theory in detail (and so can learn others in detail). Of course, some schools have quals that are not so concerned with explicit modes of knowledge but with a wide breath of knowledge (Harvard is an example, U of Ill is another). These approaches are not bad either as satisfying the requirement is usually somewhat more flexible (i.e. at Harvard you can take the qual as often as you like until you pass and they see this as an essential feature to their program). So, in part, I think the qual is not the hardest part of the Ph.D. It is only the most boring because their is form and structure--its a test. And for this test you have to take these courses and learn this material and work these problems and be able to perform at least at so and so a level within the material. So, I would say it is kind of the thing you have to do to get to the real purpose of doing the ph.d.
The next step is specializing. So, now you have to go to the "forefront" of knowledge in a particular field. I guess yours would be a certain area of mathematical physics. The test for when this is finished is the oral examination which is where you usually present a currently published paper(s) and maybe some other material to a few mathematicians. To do this you have to read a lot. I mean after the quals you may say you are at somewhere around a 1950s mathematician (or something). You have to now pick an advisor and start reading books and papers on you own to get to the present time. If you pick something like graph theory you may get their very quickly, but if instead you pick algebraic k-theory (and you don't understand some more fundamental topics) then it may take a very long time. I guess this was the point someone was making earlier about stochastic analysis. The truth is, at a young age, you have the ability to learn things very quickly when your concentration is fixed, and this time is when you really get to learn the subject you are most interested in (and presumably is why you are in graduate school). The oral presentation I am sure is intimidating and is especially for me because I often stick my foot in my mouth and overlook the obvious. But, its ok not to know something or to do something incorrectly. The main objective is to see that you know your subject.
Finally, the research aspect. To me, this is the hardest part and is probably the most difficult. I haven't any real idea what it is like. But, I do know things can go horribly wrong (e.g., someone solves your thesis problem before you do or you realize that your problem is way too hard to work on in the time that you have). Really, this should be the most exciting time though and is when you get to develop your own, original ideas.
I think the trademark for grad school as opposed to undergrad is that sometimes in grad school you are thinking about mathematics so much that it spills over into your dream time. Anyway, I would stay positive about this though.
That's basically the concept for the first year (or two, depending on the school) in most grad schools - you devote your entire life to coursework, and if you don't want to then there are plenty of other folks in your classes who are happy to get the good grades and watch you flunk out. It's not as bad as you might think, it's temporary and there's life after grad school, but while you're in the meat grinder you need to be motivated to push on regardless of how much it affects the rest of your life.