Also, I took courses on Set Theory and Topology, and fell in love with what I could garner from the experience.
You shouldn't judge math based on the first courses. I made that mistake, too. I thought that just because I took real analysis and topology and was good at it and liked it that being a mathematician was a good idea, but that turned out to be a disaster for me. I thought I was hot stuff because I was "doing real math" and my professors even seemed to agree with that, but it was a bunch of baloney. Now, of course, you may have a different opinion of the research-level stuff than I did if you get there, but I'm just saying it's really nothing like your first topology class. There's just as big of a difference between calculus and your first topology class and topology research, if not more. Also, if you are not a super-star in research, you should have a strong interest in teaching because the way the job market is, I don't think it's THAT bad, but if you study pure math, you may very easily find yourself teaching at a community college or maybe a 4-year college. So, my advice to people wanting to do pure math is to set aside some time to do some tutoring and practice public-speaking skills, unless they have a good plan to become more marketable to industry in grad school (for example, a lot of people doing numerical methods will get a lot of programming experience and maybe they could get an internship towards the end and then they would be probably be okay leaving academia). One of the big reasons it was a bit of a disaster for me was that my teaching was not well-received, and I don't enjoy it, so given that I'm not a research super-star, it's either career change or teaching-oriented stuff (I'm opting for the former). Some might say that I gave up too quickly on research and that it takes time to get good at it and not be overwhelmed by the amount of information, but they would be misconstruing my reasons for giving up on it. It's true that I now consider myself to be pretty bad at it, but even if that were just the initial hurdles of getting started, I decided that it was just too abstract and too complicated and too removed from reality, so I don't really even think it's worth being good at it anymore.
Furthermore, I've been told there is a lot of abstract math ideas in many areas of physics. In what ways and areas do abstract math and physics overlap?
Well, I think about classical mechanics in terms of manifolds and symplectic geometry and that sort of thing. So, you can view a lot of stuff through a more mathematical lens. I found it enlightening. You could try reading V I Arnold's book, Mathematical Methods of Classical Mechanics to find out more about this. Also, there's stuff like the use of fiber-bundles and Lie Groups in particle physics (but physicists have a tendency to just take what they need from these theories). Roger Penrose used some topology/geometry techniques to prove the existence of singularities in black holes.
http://en.wikipedia.org/wiki/Penrose–Hawking_singularity_theorems
He also wrote a book called Techniques of Differential Topology in Relativity.
Beyond that, stuff like string theory or loop quantum gravity is very mathematical, but it's also kind of pie-in-the-sky stuff, that's a bit out there in some ways.
What sort of careers could I hope to have that would require knowledge of math and physics in these (theoretical/abstract)ways?
Mainly just being a professor of math, physics, or engineering or something like that. In industry, the places to do that sort of thing are few and far between and very competitive. There are a few spots for it, like maybe some quant jobs, but I wouldn't count on getting one of them. The more likely route is to have to pull a massive career change maneuver and not use much of what you learned if you leave academia.
