Having gone back and looked up some things about Godel spacetime, yes, I realize I was confused on this point:Being non-orientable isn’t exactly the same thing as having closed timelike curves.
the entirety of Godel spacetime is non-orientable
Kerr spacetime violates property #2; once you've entered the non-orientable region
I found a recent paper that discusses (briefly) the fact that Godel spacetime is time orientable:the OP of the stack exchange thread is wrong when it says the diagrams it refers to are of the only non-time-orientable spacetime ever described "outside of de Sitter space with some identifications" (not sure what that refers to either). Godel spacetime, which I've already described in this thread, is another.
I would expect that Kerr spacetime is too, for similar reasons (basically, that even though there are closed timelike curves, they are in the "tangential" or "going around with the rotation" direction, so to speak, so they still have a consistent future/past orientation because the rotation in these spacetimes has a definite sense--it's a rotation going around in a definite direction).
That makes me wonder if there is any known spacetime that is an actual solution of the Einstein Field Equation but is not time orientable. As I mentioned previously, the diagram in Wald does not describe one; it's just an informal example and no corresponding solution to the EFE is given. The stack exchange thread mentions "de Sitter space with some identifications", but I have not been able to find any reference that explains what identifications.