If you are directly employed by the university (as a professor, a janitor, groundskeeper, etc.), then you would say "I work for the university." If you are performing some sort of work on the campus but are not employed by the university, you would say "I work at the university."
General Motors never has "the" in front of it. It's just General Motors, or even just GM. "I work for General Motors" is the most correct phrase, although a lot of people say "I work at General Motors", which is also acceptable.
The only time you would use "with" is if you are not an employee of GM, but you (or your company) is doing a collaborative project with them. "Right now I'm working with General Motors trying to develop an ad campaign for their new mini-van."