Graphic design is really a pretty loose term, it can be applied to many areas of design. Jobs can pay well, or it can pay very little if at all. It really depends on your talent, how much you choose to charge, the strength of your portfolio, etc. You can go to school to study yes, as I said though graphic design is a loose term so you will need to be more specific.
That is a weird question. You can make good money, and you can make crap money depending on where you work, or who your clients are and so forth.
At least where I live I can go to 4 or more well known and popular school for Graphic Design along with every other type of design out there as well.
In terms of what graphic designers do, everything. The term to me and people I know means they do everything. Me and all the graphic designers that I know do everything from business cards, logos, shirts, to anything and everything web and all the way to stuff like annual reports for companies. It's really up to you what you do. A graphic designer designs what people see in the world.