So, the horizon would be everywhere inside the gravitational wave. However, the black hole background would be a boundary. It is as if you have an event horizon, and then you have a big black hole in the background that is not your event horizon.f
If black holes absorb gravitational waves, this would allow to perform interference experiments with gravitational waves - at least in theory.
I would expect that those waves are bent close to the black hole, similar to light, so you would still see gravitational waves behind. But that is just speculation.
Yes, gravitational waves do penetrate black holes. This is an equivalent phenomenon to the event horizon fluctuations. A gravitational wave going through a black hole means that the black hole is changing its shape. On the other side, a black hole that is changing its shape for any reason emits gravitational waves. These are two views of the same phenomenon. You may imagine it as a droplet with a distorted surface.
Event horizon may go "forward" and "backward". You can define mathematicaly a sphere with a center inside the singularity. Then, some parts of the event horizon will be "above" the sphere and some will be "below". The shape may change with time. It may jump above the sphere at some point then it may submerge. In fact, it may do harmonic oscillations. The shape of the event horizon may change - the only requirement is that its total surface never decreases. The mathematics that govern the event horizon dynamics is exactly equivalent to the dynamics of a superfluid - a fluid that has no viscosity.
Of course, this process may not take away any mass from inside the black hole (classically). If some particle is caught below the event horizon, it may never come outside. Speaking bluntly, particles inside the black hole fall faster than the event horizon chasing them.
Things become interesting when we consider quantum effects. The event horizon fluctuations then may carry information about the matter that had fallen inside the black hole. So, no information is lost. All information about matter falling inside the hole is encoded in its event horizon fluctuations. All information within a volume of some object may be encoded on a surface of an event horizon of a black hole of the mass equal to the object's mass - this is the Bekenstein bound.
The fluctuations of the event horizon make up its microscopic structure. That explains why a black hole may have temperature. The event horizon surface is then proportional to the entropy. That is why it usually may only increase - and why the hole is "black" (nothing can escape it). However, as we know, entropy is our lack of knowledge about the microscopic structure of an object. If we happen to know it, it no longer increases. That explains the Hawking radiation. That means - the principle that nothing may escape a black hole is strictly equivalent to the second law of thermodynamics. However, on the microscopic scale, all processes are reversible, entropy does not increase and that is why matter may escape the black hole.
I want to state it clear: all these theses above are highly speculative. However, they appear in many theories, be it string theories or quantum gravity. Also, I let myself to recommend here all the Leonard Susskind's books, where I read all this from.
Does it really make sense to ask whether a gravitational wave penetrates the horizon?
The horizon is defined by the full geometry; splitting geometry into a propagating gravitational wave and a black hole background does not really make sense. The gravitational wave has no position w.r.t. background spacetime; only the fully spacetime is spacetime.