Could black holes be formed by highly energetic gravitational waves? Could the gravitational waves released by two merging black holes contain enough energy to produce another black hole?
 A: Colliding gravitational waves can indeed form black holes, but the conditions for doing so are fairly strict. As it happens this has just been discussed in the preprint Black Hole Formation from the Collision of Plane-Fronted Gravitational Waves.
But it's exceedingly unlikely this would every happen. The gravitational waves near merging black holes will be spherical not plane waves and it's unlikely they could concentrate enough energy to create a black hole. The waves will become increasingly planar as the move away from their source, but of course their intensity will fall off with distance squared so far from the black holes they certainly won't have enough energy to form a black hole.
The tl;dr is that yes it's theoretically possible but in practice it's hard to imagine circumstances where it would actually happen.
A: In theory, yes, any concentration of energy that is suitably high will form a black hole. In particular, according to general relativity, if any energy is concentrated within about a volume equal to its own Schwarzschild radius or less, it will form a black hole, and given that gravitational waves transmit energy there is no reason this would not apply to them as well. It would effectively be a gravitational version of a so-called "kugelblitz" - a a black hole formed by a similarly-intense and concentrated convergence of electromagnetic radiation. Kugelblitzes are not something that has ever been observed, I believe, and may not have occurred anywhere, since it is very difficult to concentrate such highly mobile forms of energy.
And thus to answer the question here, following on that last point - no it could not, because the gravitational waves are radiating outward, so their concentration is decreasing. The energy of the waves comes from the orbital energy of the black holes - if that were enough to form a black hole then that would effectively mean the black holes had merged already, as they'd now be inside one larger horizon due to it (and thus also to a single large core, presumably) since it's the total energy and momentum in the system that give rise to the gravitational forces (as well as the system's total mass - note that this added mass doesn't belong to each black hole individually but the system collectively).
