Wouldn't it require something very obvious and easily observable to be causing the curve to exist, like a wormhole?
One may, for example, take a flat Minkowski space-time and impose periodic boundary conditions creating a model with CTC. How would one go about demonstrating or refuting a CTC in such a model when locally it's the same as one without a CTC?
As this example seems to show, CTC is only an artifact of bad event labeling.
No, it isn't; the two spacetimes you describe have different topologies and are therefore different manifolds.
You would have to find a way of probing the global topology of the spacetime;
My use of "locally the same" I believe is correct if I recall my topology classes.
I know of no nonlocal measurements that are not in some way comprised of local ones.