The axiom I presented for Hyperbolic Geometry is the correct one. It can be used to prove that there are infinitely many parallel lines to each line, going through a point not on it.

But you are correct, my statements were a bit irrelevant. And your last conjecture seems to be true, to me. Mathematics, as far as I know it, is not really interested in what exists in the real world, and what doesn't. It does interest itself in the logical consequences of properties of objects. An Applied Mathematician can then say "Lo, for this Physical/Engineering-related/Literary(?) problem can be translated into yon Mathematical object, where it can be dealt with by the tools of Mathematics."