No, that is a mistaken visualization that comes from embedding the donut in a higher space. No embedding is necessary, nor does anyone think the universe is embedded in anything larger.
Think about PacMan, or the old Asteroids game, where going off one end of the screen would put you on the other side. That's a donut. (The 4 corners of the screen make up the single hole*.) Which "side" is the inside? The question doesn't make sense.
*Edit: as rightly pointed out below, the location of the hole is an arbitrary choice that comes from trying to map the space to a sphere, and does not actually exist anywhere in the space itself.
An interesting experiment is this: imagine yourself existing in the space, which is otherwise empty. PacMan alone in the middle of the screen. Throw a stretchy rope to yourself, horizontally or vertically, catch the other end, and tie it together. Then walk around the space without turning the rope at all. Notice that no matter how you walk around, the rope will always be the same length. Now imagine the same thing on the surface of a sphere. Walking around makes the rope larger or smaller, and there's always a point you can walk to where the rope will completely collapse to a single point.
Another way to think about it is that in order to be on either "side" of the flat sheet, you've implicitly introduced depth, and it's not really two-dimensional anymore.
If you were in a two-dimensional universe, you wouldn't be on the paper, you would be a patch of the paper.
I don't understand why the 4 corners of the screen make up the single hole. You could scroll the screen by 1 "square" which would change the corners which makes me feel there is nothing special about the original four corners.
You're right that the original 4 corners are arbitrary. The hole isn't physically present in the actual space. In fact the word "hole" comes from visualizing the space embedded in a higher space, which we know is not necessary and invites misconceptions. So let's call it a discontinuity.
The discontinuity shows up when you try to continuously map the space to the surface of a sphere. You can almost do it, except for one point. Different nearly-continuous maps have a different point of discontinuity -- it's basically your choice when doing the mapping. I think the 4 corners feels like a natural place for the discontinuity when I visualize that mapping -- and scrolling feels like selecting a different mapping -- but indeed it could be any point in the space if you visualize that mapping differently.
Yeah the corner can correspond to any one point on the torus. They are all the same point, but other than that there's nothing really interesting about the corner(s).
The edges are more interesting, two of them go around the 'hole' of the 'donut' and the other two wrap 'around' the 'donut' itself (i.e. around the dough if it was an actual american style donut). There's no way to tell which is which.
These edges have the interesting property that you can't shrink them to a point (compared to say a loop on a globe which you can make smaller until its a single point). Except when the donut is not hollow in that case one of the loops becomes contractible, turning the space into the equivalent of a circle.
IIRC PacMan was not a torus. Asteroids was, but not PacMan.
Edit: Just looked it up. We're both right, sort of. There were warp tunnels on the sides, but not the top and bottom. PacMan was a semi-torus. IOW a cylinder.
