Some hints below that maybe will help you (otherwise please come back to me).

You're definitely right here, it sounds weird. I don't know much or anything about General relativity. But I was thinking since the universe was made of space and time, how is it a flat surface (where does the depth comes from, if it was a surface?).

Everything likes in the definition of flatness (and of surface by the way). As I said in the post, a cylinder is a flat surface. It is all a matter of shooting parallel lasers and have them never crossing each other.

What defines the universe? I mean if the universe was some surface, it should be contained in some other material that has a volume, since space itself is a quantity that has volume.

We have the observable universe that is well defined. For the rest, we can speculate anything as it is impossible to test it and verify it.