You seem like an kid who needs some real career guidance. I don't know if you have potential or not, but listening to the guys on ET is about the dumbest thing you can do if you are actually considering a real career in the investment business (not that there's much of a business left these days). Most of the guys here never had a shot with the big boys.
1. This may seem like a weird first question, but bear with me - how's the math in trading? Does the career require a genius mathematical mind, able to do long division in a quick mental flash, or is a decent, basic grounding good enough?
A lot of roles won't need any math beyond high school math. The jobs that do will require math you've never heard of and it certainly has nothing to do with how fast you can calculate in your head. Depends on your career path.
2. What kind of education is needed? Do you have to graduate from business school, is a position as a big broker at Goldman Sachs only available to Ivy Leagues? And what kind of courses should I be looking for in High School, then in College?
A good school (not necessarily an ivy) is necessary. It gets you in the door and it signals that you've put up enough bullshit to thump your fellow graduates who are all talk and no game. In highschool, do what you can to get into the best school you can. In college, look for internships and other ways to further one-up your peers.
3. As a stock trader, are you allowed to invest yourself? It seems like, if you were the one ordering the buying or selling, it'd be illegal for you to take part in it.
Yes and no. Yes in that you theoretically can. No in that there will be all sorts of personal trading rules and procedures (like you can't sell a stock you bought within 60-90 days, or you can't buy or sell certain restricted stocks). You certainly can't day trade.
4. How easy is it to climb the ladder in these kinds of careers? And where do you start, on average?
Not hard if you want a comfy 200k-300k life (inflation adjusted). Cut-throat otherwise.