Verified answer

As part of their training, a computer engineer will learn a little bit of programming, but it's probably not quite enough to develop games. You could probably make simple games (arcade-style) or you could use some hand-holding tools like Game Maker, but nothing terribly complex. You'd need to read up some more on things like software design to make more complex games.

Technically yes, although that's more along the lines of software engineering rather than computer engineering.

Anyone can be a game developer, you don't have to do it as a career, it can be a hobby.