Pacman's Adventures in Curved Space

Bengt and his friend Alban are playing Pacman on Bengt's new (and very expensive) calculator. The screen is 1x1 dm. Alban is, unlike Bengt, younger than Pacman, and he has played a lot of computer games. Neither of them knows much about the theory of relativity, but Bengt has a habit of discussing things that he doesn't understand.
"This is a bit like the theory of relativity." says Bengt.
"Really? How?" says Alban.
"Well, when Pacman leaves through the hole in the wall in the middle of the top edge of the screen, he immediately reappears at the corresponding hole at the bottom, and the other way around."
"You mean that he has moved faster than light?"
"No, that's not what I mean. Don't try to tell me what I mean. No one but me knows what I mean."
"True... so, what do you mean?"
"That he comes back, it's like if space is curved. As if he lived on a cylinder. The Y coordinate repeats itself; it's a cyclic coordinate."
"Uh... that's not what 'cyclic coordinate' means. It means a coordinate which the lagrangian is independent of, or something like that."
"I don't care. If a coordinate repeats itself cyclically, I will call it a cyclic coordinate. And apart from cyclic coordinates, there are also limited coordinates, which end, and adifatic coordinates, which neither end nor repeat themselves. On a cylinder, one coordinate is cyclic, and the other is limited."
"Okay, if you say so..."
"If there were holes on the left and right edges as well, so he could go around in the X coordinate as well, Pacman might get the impression that the world was round. Spherical. Or that he was living on on a doughnut. One might also imagine parallel universes, in the shape of doughnuts linked as a chain. I call the theory "the catananese doughnut theory".
"Blast! Now I died again. Could you please stop talking geometry? I'm trying to concentrate."
"No I won't. It is also possible to let him leave and reappear not only in the middle, but anywhere on the top or bottom edge, and we can imagine that the screen is infinite in the X dimension, so that that coordinate is adifatic. Then it would be like an infinitely long cylinder, and if Pacman seems to move on a straight line on the screen, he is in fact moving in a spiral on the cylinder surface, and depending on his direction the time it takes to get back to the same Y position varies. It's just like in string theory."
"Sure, sure, a spiral, on a doughnut..."
"No, that wouldn't work. One can also imagine that he can leave anywhere on any edge, but it will become neither a sphere nor a doughnut."
"Cut it out, I'm getting a headache."
"Okay. Beer?"
"Yes please."

1. If Pacman lives on a finite cylinder, what is the surface area and volume?
   
   
2. What if it is a sphere? What is the surface area and volume then?
   
   
3. Since the screen is a square, is it really possible that he lives on a doughnut? How big is the hole?
   
   
4. If Pacman is spiralling along an infinite cylinder, what is the time until he returns to the same Y coordinate, as a function of the angle between his movement and the X axis?
   
   
5. If he can leave anywhere on any edge, under what conditions will he ever come back to the same point (without turning)? How far will he then have moved?
   
   
6. If he can leave anywhere on any edge, why is it impossible that he is on a sphere or a doughnut?
   
   
7. In that case, if he is moving on the surface of some shape, what kind of shape is it? How many dimensions does it have?
   
   
8. What is it's volume? (That is, the equivalent of volume in however many dimensions it has.)