I found this Matt Parker lecture provided some good tools for getting your head around the fact that extra dimensions really do let you go around things, not through: https://youtu.be/1wAaI_6b9JE?t=2299
(the whole thing is worth watching, but the bit about higher dimensional visualization starts there)
I'll save people the effort and point out that the only relevant things (to the given questions) the entertaining video shows are a proper klein bottle and the nature of projection into 3D. In 4D, klein bottles don't self-intersect, just as mobius strips don't self-intersect.
It does not cover the question of putting oneself inside a wall or how to go around 4d objects, and the projections show detail that a 4d person would not see.
(the whole thing is worth watching, but the bit about higher dimensional visualization starts there)