On the question you mentioned, a commentator said, "astrophysicists would be very surprised to find a nonrotating black hole in nature". And the event horizon of a rotating black hole isn't actually going to be spherical.
Anyways, the relaxation to an oblate shape might be quick. Now, this is a messy business. There have been approximation and numerical methods used to analyse the merger of two black holes. These are way over my head, but Figure 2 of Binary black hole mergers in Physics Today (2011) shows the ring-down time being a hundred or so times $GM/c^3$. ($GM/c^3$ was the characteristic time mentioned in the comments to the other question, so this is in agreement with what was said there.)
For a solar mass black hole, the characteristic time is about 5 microseconds. The supermassive black hole at the centre of our galaxy is thought to be about 4 million solar masses, so the time would be about 20 seconds. So the ring-down time even for that monster would be only about 2000 seconds, or let's say half an hour.
That said, this only models how long it takes huge distortions to relax to small distortions. It's not clear to me that small distortions have as fast a relaxation time to even smaller distortions. More precisely, I don't see why the decay would be exponential. Again, it's over my head. [Maybe this should be another question.]
You also asked if there could be other periodic disturbances of the horizon. Technically, no, any disturbance would be subject to some damping, because it would have to produce gravitational radiation. If an object were orbiting the black hole, for example, that would have to distort the event horizon as it passed over it, while its orbit would decay via radiation. But the power radiated doesn't scale linearly with mass of the orbiting body. For very small disturbances, it could take a very long time, and you could have an almost periodic scenario. (In the limit, test particles have stable orbits and produce no distortion of the horizon.)