The slowness of black hole evaporation really is astonishing. Even though I already understood stood as much, I was still taken aback by how small the correction to the event horizon size because of it turns out to be. I figured it would at least be, like, a micrometer or something, which you could say is significant enough to be real but still small enough to ignore. Less than a Planck length is ridiculous.

Aside: a similar thing happens with how solar wind modifies planetary orbits. The wind constantly pushes outwards, so shouldn't the orbits expand? Again it's one of those things that turns out to be negligible. Since it pushes radially outward, it doesn't give the planet orbital energy. Instead, the pressure from solar wind acts like a constant centrifugal force, which is like slightly decreasing the Sun's effective gravitational force. This makes the circular orbit at a given orbital speed slightly larger than you'd expect, but it does not grow over time. For Earth, it raises the orbit by about 1 micrometer.

Let's say that Alcubierre drive shipships are possible. Could they go through a black hole by bending away the event horizon?

If the ship could go faster than light, absolutely. Not by bending the horizon away, but by following a spacelike path, which exits its own future light cone. At the event horizon, the outgoing edge of the light cone is parallel to the horizon, and tilted more inward the further in you go, so a greater FTL speed is needed to get closer to the center and emerge again.

*When* would the ship emerge, according to an external observer, I'm not sure has an invariant answer. But this scenario of course introduces all sorts of paradoxes even without a black hole involved.

I think you're asking though if a slower than light ship using a warp bubble could escape by having some effect on the horizon. My current thinking leans against it, for probably an unintuitive reason. The reason is that the curvature of spacetime near the horizon is small, and unrelated to the existence of the horizon. If we imagine an arbitrarily large black hole and a ship a small distance inside it, the local environment of the ship is no different than being in flat spacetime far from any black hole at all. But the horizon now behaves the same way as a receding flash of light. Even if you accelerate to arbitrarily large fractions of the speed of light, whether by using rockets or by a slower than light warp bubble, that horizon remains ahead of you (even getting further away), in the same way as if you were trying to catch up to a light flash.

That answer might be confusing, because isn't curvature the very thing that "causes gravity" by "telling matter how to move" in the first place? That's right, but in the sense of telling how nearby particles will accelerate relative to one another (tidal forces). Otherwise, the only thing any particle does at the local level is "move straight", i.e. along a geodesic, in spacetime. It is the global geometry rather than local curvature that constrains geodesics to move more inward.

at 0:50 we see an extreme opposite warping. Does this imply that it's possible to ride these waves for a split second to achieve faster-than-light travel as seen for a distant observer?

This is pretty neat. The short answer is no, but

there's a lot of information to digest here, so let's cover each part of what the video is showing.
The colors represent the magnitude of time dilation. Arrows represent the direction and magnitude of acceleration (as seen by a stationary observer at that location). The height of the sheet represents a measure of the warping of space. What measure, exactly? I'm pretty sure it is the spatial part of the metric, which is the most common way to create this visual (also called an embedding diagram) for a black hole.
The Schwarzchild metric for a static, non-spinning black hole in the time and radial directions is
The first term in parentheses gives you the time dilation, while the second term tells you how much the space has been distorted. This is not the curvature in the sense of tidal forces, but rather in the sense of how much the proper distance (which you would measure with a ruler) between two points in the radial direction is different from what you would expect if you were in flat spacetime. See my earlier post here for more explanation and a thought experiment you could imagine doing around the Earth, in order to interpret this type of spatial distortion.
For a single, non-spinning black hole, this distortion changes with distance in the same way as the time dilation does (those two metric components are just negative inverses of each other), and the spatial distortion is typically plotted with a minus sign so that it looks like a funnel. For a black hole merger, some regions get the spatial component distorted the other way (upwards in the diagram), even while the time dilation is still very strong so the colors are red. That upward distortion isn't like a region of antigravity or a white hole though -- we can see the arrows still point towards the black hole through this region, and have very high magnitude. A test particle released in this region would quickly fall into the merged black holes. But the warping of spacetime is obviously very different from normal. If the usual downward curve means that there is more distance between nearby points in space than you expected, an upward curve means there is less. Space is, in a sense, more compressed in the radial direction than you would have expected. (I expect it is more stretched out in the perpendicular directions.) The large magnitude also implies large tidal forces -- this is where the gravitational waves have the greatest effect.
It's fun to do a quick calculation to see how violent the gravitational waves are this close to a merger. The fractional amount something is stretched and squeezed by the wave (called the strain) is inversely proportional to the distance from the wave source (whereas the energy of the wave drops off with the inverse square law.) At Earth, the strain was about 1 part in 10^21, and the source was about 400 Mpc away (give or take about 150 Mpc.) Each black hole was about 30 solar masses, or about 180 km in diameter.

If we ignore the fact that the regular tidal forces are already lethal this close to black holes of this size, the gravitational waves are probably lethal, as well. At a distance of 1000km (just outside of the final orbit they made), the waves carried a strain of about 1 part in 100. That would stretch and squeeze your body in alternating directions by s

everal centimeters, multiple times in much less than a second. I'm not sure what that would feel like, but I doubt it would be pleasant, or even survivable!