Near the end of the movie where they expelled the brown dwarf star piece. The moon should have been the one that's expelled while the star piece stayed. The mass of the star is twice of the Earth's. The moon is 1/100 the mass of the Earth or 1/200 of the star. The end of the world will still happen but without the moon. Perhaps they miscalculated the star's mass, and the actual mass is less than the moon's.
Every time we see the moon after the impact it is surrounded by debris floating in space. If the moon and dwarf had enough gravity to lift cars and people from the surface of the earth, it would also be able to pull the debris back to its surface.
At least twice in the movie a character refers to the moon's magnetic field. The Earth's magnetic field is produced by convection currents of iron-laden molten rock at the Earth's core. The moon has a solid core and cannot generate a magnetic field similar to Earth's. There are some isolated fields on the moon's surface, but these are very weak. Like the moon, Mars also has a solid core and no magnetic field. A compass would be useless on either the moon or Mars.
There is no way that the moon would resume its original orbit after the perturbations it went through from the original impact the subsequent nuclear impacts and the final blast.
When the Moon passed close enough to Earth to cause zero gravity, it would have either altered Earth's orbit around the sun or split it in two.