I'm not a game programmer but I've read numerous websites on how to get into the video game industry and have read a book on it (becoming a game programmer is my dream). The obvious choice of a degree when going for a game programming position is computer science, though it is definitely not required. According to sources I've read, a major part of what will get you a job is knowing people (which can be done by attending game conferences, getting your foot in the door through a game testing position, and so on), industry experience, a real passion for making games, and a really good demo reel showing that you know how to make games and you make them in your spare time. If you want more details go to
www.monster.com and search for game programming jobs, you'll find a lot of information there.
A place I'd highly recommend you visit is
www.gamedev.net. It's pretty much the top site out there for aspiring game programmers/artists/whatever. The forums are loaded with smart people (some who
are game programmers). There's a section on math and physics over there but the real emphasis on that forum is on programming, particularly in C/C++.
You might also want to check out
www.gamasutra.com too. There's a lot of cool information there.
Within those two sites you will find links that lead you to even more sites with even more information so I advise you snoop around if you really want to get to know about game programming. Believe me there is
a lot of information online about the things you asked for. I used to know of a site written by some game programmer that went into a lot of detail explaining what the typical day is like, but I've lost the link unfortunately. If I happen to come across it again I'll post it here.