It would certainly looks like you traveled faster than light. But you didn't actually, you just took a "shortcut" as it were.

There are no shortcuts. The straight line connecting 2 points in 3d space is shorter than any path connecting the points that goes through a higher dimension. The same applies to 2d compared to 3d. If you have 2 points on a sheet of paper, the shortest path is on the paper. Any path that went "off" the paper into the 3rd dimension would be longer, so no "shortcuts".

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).

We're talking about a game. We know exactly what geometry the game world has.

No we are not. He asked about the speed of light.

And the game world has some super weird discrete geometry anyway.