I have to start by saying that, from a scientific perspective, there are a couple of really major problems with this scenario, before we even get to the physics of the energy required to make such a crater.
The first is that looking at the picture above, the moon is blocking out the sun, and yet, a quarter of the surface is illuminated. This is geometrically impossible; the sun illuminates the surface of the moon that faces the sun - there’s no way for sunlight to be reflected off the surface facing away from the sun.
Secondly, the way the crater appears, it looks like the moon is a flat disk that had a notch bitten out of it, like a cookie. The same is true if you watch the clip of the actual explosion, where they shoot a hole in the horizon so that they can see the Earth. The problem with this is that the moon is a sphere, not a disk, so any divot like that would have to be a cylindrical trench in the surface of the moon, not a spherical one like normal craters make. This cylindrical hole would only appear as a semicircular divot from a few positions in space, so that final shot would not see the chunk taken out of the moon as so clean a semicircle, since it’s looking from a different angle than the original firing.
To figure out what kind of damage the creation of a crater like this would do to the moon, I had to make a couple of simplifying assumptions. The major one was that the depth of the hole we see in the final shot is the depth of the crater, and that the width was equal to the width of the crater. I also assumed that the crater was round, and not a tunnel, since the explosion looked round in the clip. (It’s really hard to make cylindrical holes without a lot of prior planning.) With this information, I could calculate the volume of the crater, and using the average density of the moon, I can find the mass of the moon that had to be removed to make the crater. The above picture of the moon was very useful for this calculation, since I could measure the scale of the crater relative to the entire moon, which allowed me to work out that the crater is 1390 km wide, and 463 km deep. With the assumption of the crater being spherical, and the average density of the moon of 3.34 grams per cubic centimeter, this leads us to a total mass removed from the moon of 1.347 x 10^21 kg. This seems like a huge amount of matter (and it is), but since the entire moon weighs 7.348 x 10^22 kg, this is only a removal of about two percent of the total mass of the moon.
Since the final shot of the moon is some time after the explosion, and there’s not a lot of debris hanging around, I worked out the amount of energy required to both lift that much mass, and the amount required to fling that much mass permanently away from the moon. It’s a lot of energy. In fact, it’s more energy than could be generated by the entire world’s nuclear stockpile detonating at once, by a factor of a billion (10^9) - and that’s just to lift it. To permanently remove the mass, the current world nuclear stockpile is a factor of a quadrillion (10^15) too weak. The Götterdämmerung only fires two bullets - we can safely say that these bullets are physically impossible.
As far as whether the moon would eventually become a sphere again, the answer here is yes, it would, but it would do so in a way that’s very different from what’s depicted in the film. The film essentially depicts this massive detonation as resulting in what’s called a “simple” crater. A simple crater is just a bowl shaped cavity on the surface of a planet or moon, and the moon has plenty of them already. But when a large amount of energy is injected into the surface of a planet, a lot of the energy is lost into heating up the rock; that heat will temporarily liquify the surface. The surface will ripple, and then cool down, resulting in a series of rings and (often) a central peak - this is called a complex crater. This liquefaction and rippling means that the craters are not as deep as they would be if every crater was a simple one. The bottom of the crater, being a liquid, rebounds up towards what would have been the surface, and a lot of the material that could have been thrown out of the crater would stay on the surface as a liquid. These events are still quite destructive, and a large amount of material would be thrown out of the crater, but the moon would never have become quite as non-circular as it was depicted.
One of Saturn’s moons, Mimas, has a really massive complex crater on its surface that earned it the nickname the Death Star Moon. Mimas was thought to have been nearly shattered by this impact, and has a bunch of what is technically called “weird terrain” on the exact opposite side of the moon. Weird terrain is where the seismic shock waves from the impact all meet up and collide on the opposite side of the object, releasing their energy. This fractures the surface in a particularly chaotic way, and can trigger volcanism if the planet or moon has magma hanging around under the surface. Mimas doesn’t, and our moon only has a small molten core, but an impact like this would certainly jumble the surface at the antipode of the moon.
The moon’s orbit would be remarkably not affected by this. The material thrown out (as shown in the film) from the moon is one fiftieth the mass of the remaining moon - and the laws of momentum tells us that an object fifty times larger is fifty times harder to move. Assuming that the material blown off the moon reaches escape velocity, the rest of the moon will receive a kick of 44.5 m/s. If the material doesn’t make it to escape velocity, the kick will be weaker. The moon’s average orbital speed is a touch over 1 km/s, which means that this additional velocity is only an alteration of 4% to its speed. However, we have to consider the geometry of the situation here - if the moon had been hit along the direction of travel, it would gain or lose speed in its orbit around the earth, and that might change the distance of its orbit by a small amount. But the hole is in the “top” of the moon, which means that the velocity boost to the moon is pointed “down”. This won’t change the distance of the moon’s orbit, but will make the orbit slightly more tilted. Only slightly, though, since the velocity kick still isn’t very strong.
Overall, this crater would not dramatically change the orbit of the moon, though it would make it more tilted, and if this had happened in the real universe, a lot more of the energy would have gone into melting the surface of the moon, and a lot less into making a crater.
Something here unclear, or have your own question? Feel free to ask!