Good question. From the point of view of a “clock” maintained at either black hole singularity they will. From the point of view of a remote observer watching the merger it will take an infinite amount of time.

“What happens to the event horizons during the early stages of the merger process? Would they extend outwards towards the other mass?”

The event horizons are not physical objects so there should be no effects similar to what happens when two physical objects merge (e.g., turbulence, etc.).

“If orbiting black holes generate gravity waves and waves are a form of energy, wouldn’t those gravity waves be getting their energy directly from the masses of the black holes? Is this a way for a black hole to lose mass, like Hawking radiation?”

It’s my understanding that the energy contained in the gravitational waves comes from the initial kinectic energy of the two black holes.

“If so, is this a significant energy loss to the black holes?”

Probably not, in this case the kinetic energy of the smaller of the black holes is 7.2E50 joules but its rest energy is 3.6E54, 5000 times greater.

“Would the merged black hole rotate, having gotten a big kick from the orbital motions of the original pair?”

They are probably both rotation to begin with. After the merger the resulting angular momentum vector is the vector sum of the initial angular momentum vectors (using the parallelogram rule).

“I seem to recall that a rotating hole had a singularity that was a ring, rather than a point (or was it a double event horizon?).”

I think that’s probably true. Of course any attempts to go inside the event horizon and verify this are likely to be fruitless.

“Wasn’t there some idea that an energetic phenomenon (Gamma Ray Bursts?) was the result of black hole mergers? Or was it simply black holes consuming large masses like stars?”

GRBs are thought to be coming from massive stars during black hole formation, not from swallowing stars although that probably would be an energetic event also.

Eki-eki-eki-*what-is-this-last-sound?*