Your thought experiments are interesting and not all that easy to answer. You are now bringing up the constraint of a "rigid body", which is an idealization that does not exist in reality.
I think davebee is correct to bring up the classical Newtonian physics limits of your first example. But if we take your example and add the rigid body and rigid string constraints then Newtonian physics will predict that the wave propagation velocity will exceed that of light. So we need Einstein's equations to talk further.
So, you really can't have a rigid body, once you start using Einstein's equations because, as you point out, a rigid body implies infinite speed of information transfer. You have also brought up the issue of very strong gravitational fields with a black hole, so not only is the object not rigid, but space itself is not rigid and can warp into the time dimension.
A simple (cop-out) answer is that you have to solve Einstein's field equations to calculate a real answer, and the real question would not have rigid bodies, nor rigid space/time dimensions as constraints.
