What could prevent a warp drive from entering and exiting an event horizon? Remember that in GR the region near an event horizon doesn't have particularly strong curvature. The tidal forces near a supermassive black hole will be small, so the spacetime around the warp drive will be nearly flat...
The key concept you're looking for is hydrostatic equilibrium. If a celestial body is massive enough, its gravity will cause it to assume the optimal shape which balances gravity and pressure. This is a spheroid.
Suppose Alcubierre warp drives or some alternative form of warp drive is possible. Then surely a warp drive could enter the event horizon of a black hole, observe the interior, and exit the horizon, returning with information from within the black hole. What about black holes never visited by a...