Each ship has its own mass and thrust to move that mass, yes they are simply turning that thrust off when the engines are destroyed, the broken ship is no longer able to push against the moons gravity and so it falls, one could also argue that because these moons are quite small, about 1 quarter the size of our own moon, or 1 in 10 of earth so these objects don't really hit the surface that hard.
But that's a straw argument.
It does partly answers your other question, these Moons do have realistic physics, they have a gravitational pull calculated from the centre out, the further way you are the weaker the pull, what's more once inside the celestial objects gravitational pull you are in its orbit, all these objects rotate around the Stanton Star and all the Moons of Crusader (which is what they are) rotate with Crusader, the red gas giant in the middle. so once in orbit around what was in that case Daymar you rotate with Daymar if you turn off your thrusters, if you watch the video it again you will hear him say the body of the falling ship was moving, it was not going straight down.
Having said all of that this is not easy, not by a very long way, if it was everyone would be doing it, in fact this is a first.
PS: as 'rocky' celestial objects the Earth its Moon are actually quite large in the universe.
Alpha 3.0 Time Laps....