IIRC the group velocity in a medium can also exceed c when you send pulses through certain nonlinear media. Basically, the "lump" of the pulse can appear to "exit before it entered," but what really happens is that the tiny tip of the pulse has already gone through at <= c, and starts growing into the body at the exit before it has completely shrank at the entrance because of the nonlinearity. So it is actually the information/"signal" velocity (the tip of the signal) that cannot exceed c.

Are there interesting relativistic effects related to this other than this Cherenkov radiation?

To be more perfectly precise, it is only in a vacuum with an infinitely repeating regular signal that "phase velocity of light" = "group velocity of light".

The deviation caused by uncertainty in the package arrival time is typically extremely short, but not zero, for finite signals.

The actual "entity" that is restricted to be <=c is "information velocity". An infinitely repeating pattern would have travel at (phase velocity)=c ... but would ironically transmit no actual information (the bit value is restricted to 1, not 0 or 1).