To give a very general answer - Schwarzschild spacetime relies on certain conditions that need to be in place for this particular geometry to arise. It is static, stationery, spherically symmetric, and asymptotically flat (ie there are no other distant sources of gravity). If any of these conditions is violated, we are no longer dealing with Schwarzschild spacetime, but something more complicated.
In principle, yes. But remember, a Schwarzschild BH is stationery and relies on an otherwise empty universe, meaning it doesn’t permit any changes - so you can’t have anything falling into it. If you add even as much as a single particle falling in, it’s no longer truly a Schwarzschild BH, but some other geometry.
Yes and yes. But again, this wouldn’t be a Schwarzschild BH any longer.
That’s a really good question! I presume you mean a gravitational wave. You can certainly embed a BH into a background gravitational wave field. The result would be something pretty complicated. I don’t know for sure just exactly what would happen, because, since GR is a non-linear theory, metrics don’t just add - you’d have to actually derive an entirely new solution for this scenario, which is likely only possible with numerical methods.
I can make an educated guess though - given the right wavelengths for your gravitational radiation, the event horizon of your BH would begin to oscillate and ‘vibrate’ (like a bell) and eventually achieve a state of resonance with the external wave field. But this also means that the BH itself becomes a source of gravitational radiation - so it would essentially reflect some of the radiation back out. I don’t know if it would re-radiate all of the energy, or absorb some of it and grow in mass; one would have to run the numbers to find out.
What’s more, the re-radiated waves will interfere with the incoming background waves in complicated non-linear ways, changing the wave field in ways that I can’t predict here now.
And to go even further - if you were to ‘turn off’ the external wave field somehow, the BH will slowly ‘ring down’ like a bell, and eventually become stationery; however, the surrounding spacetime will remain permanently altered by all these waves having gone through it. It’s called the gravitational memory effect.
This is a really complicated scenario, but very interesting.
Yes, the event horizon will deform and ‘bulge out’ - this happens, for example, when two BH approach one another and merge.
No, because spacetime inside the horizon is empty (assuming no in-falling material), so there’s nothing there to experience stresses.
Schwarzschild spacetime is always spherically symmetric. If it doesn’t have this symmetry, then it will be a different kind of geometry.
Yes.
No, it wouldn’t be spherical, and thus it wouldn’t be a Schwarzschild BH any longer. Schwarzschild geometry requires spherical symmetry.