Why doesn't gravity couple to spin and how would we prove it? Spin is always explained as a sort of "intrinsic" angular momentum, and orbital angular momentum very clearly affects gravity via gravitomagnetic effects (frame dragging)
If the analogy between spin and OAM holds, why is there no incorporation of spin into the EFE? I know about Einstein-Cartan theory, but how would you test the differences?
I'm also assuming if spin is involved the universality of free fall is broken by a sufficiently spinning mass: is this the only way to find if spin actually influences gravity?
 A: The possibility most often discussed is that intrinsic spin couples to torsion. This is what happens in the Einstin-Cartan theory, and is what is being searched for in experiments such as the ones done by the Eot-Wash group at UW using a spin-polarized torsion pendulum. A positive result from such an experiment would prove that gravity does couple to spin in this sense. There is no way to prove that it doesn't couple, since a null result can only put an upper limit on the phenomenon.
The "why" could just be that God gets to pick the laws of physics. Or you could say that you find the equivalence principle to be very aesthetically attractive, in which case torsion is ruled out.

I'm also assuming if spin is involved the universality of free fall is broken by a sufficiently spinning mass: is this the only way to find if spin actually influences gravity?

This is not a violation of the e.p., nor is it a demonstration that gravity couples to spin in the sense that people normally mean. According to GR, this effect goes to zero in the limit of small test particles, provided that we assume an appropriate energy condition: Ehlers and Geroch, http://arxiv.org/abs/gr-qc/0309074v1 That limiting behavior is considered to be enough to satisfy the e.p.
