John,
"Let's not call it space, let's call it an unbounded frame of reference."
I'm afraid that's what space is. Add boundary conditions and you can call it geometry.
"It has no physical markers. You are on a merry-go-round, or rather the spacestation from 2001 A Space Odyssey. Presumably you would be able to tell how fast it is spinning relative to that frame, by the G forces, not by any reference to outside points."
No you couldn't. This was the whole point of Einstein's elevator gedanken experiment -- that one cannot determine whether one, colloquially speaking, is being pulled up or pushed down, in the absence of an external reference.
"So what is that frame, if it has virtually no other physical property than what the spin is relative to? What is this frame derived from? Algebra? Geometry? Space?"
Pick one, or all three. Or a method yet to be invented. What is clear though, John, is that you are innocent of how classical physics works. I promise that if you study it, you will be rewarded with insights that not only answer your questions, but lead you right to the edge of the cliff overlooking the truly deep questions.
Tom