Imagine a strip of paper bent into a U. You can travel from one end of the strip to the other through 3D space with a much shorter distance than along the entire paper.
Sure, but that's not what we're talking about here. We're talking about Euclidean 4D space projected orthogonally onto Euclidean 3D slices, not some nonstandard projection onto some weird 3D surface embedded in 4D space.
We don't really know that. Let's suppose the universe was really 4D. We don't currently know if the universe is flat or not. A curved universe can be both unbounded and finite which would be very elegant.
Einstein proposed a test to find out. I'm hoping this test will eventually be Gravity Probe D (Probe C will be a repeat of the failed Probe B).