Exactly! the resulting object would be homeomorphic to a torus. But you have the additional 'sew' operation going on here, so it's not the same as just punching a hole in a sphere. The reason I brought this up is because while you and I may know what we are talking about, the OP might not, and thus might get confused about this.
It wasn't clear from context at all. Maybe I'm just an idiot, but I thought you were talking about solid objects, not surfaces.
Actually, you won't find the term 'doughnut' used much in math books. But still, people in math circles do use the term, and they usually take it to mean a 'filled' torus. Also, you used the word 'sphere' correctly. The solid interior would be called a ball, not a sphere.
About the string-theory part, I won't comment. I don't understand cohomology yet (even though I'm trying to!) so I don't think I'm qualified to express an opinion about string theory.
