Is space bent or does it simply appear to be bent, depending on the observer?
Mass warps space. This is demonstrated in many ways, but gravitational lensing is one observed phenomena which exactly matches predictions by General Relativity.
Does time move more slowly, or does it simply appear to move more slowly, depending on the observer?
Clocks in a higher gravitational field show a change which exactly matches the changes predicted by General Relativity.
If you travel in a straight line indefinitely, will you eventually return to your origin?
I don't know. I've lost track of which cosmological topology is currently on top of the probability list, but I think the most recent answer is "no" based on the accelerating expansion which was observed back in the late 90's.