Within only a couple of minutes of the explosion which blasts the moon out of orbit the moon quickly begins to decelerate. Since the moon was moving through the vacuum of empty space there would be nothing - not even an atmosphere - to allow for any deceleration to take place.
A tremendous explosion on "the dark side" of the Moon supposedly throws the Moon out of orbit and into deep space. There really isn't a "dark side" to the Moon. Though it is true that only one side of the Moon ever faces the Earth, all parts of the Moon get sunlight as it orbits Earth every month. Even if they meant the "far side" of the Moon, the situation is no better. If they actually meant "dark" side (ie: the side facing away from the Sun at that moment) then the explosion would have flung the Moon into the Sun. If they meant to say "far" side, then the explosion would have flung the Moon into the Earth.
There are Earth TV broadcasts after the moon has left orbit stating that Earth is recovering from the resulting devastation. Even if this was possible, Earth would still not survive long. Without the moon's tidal forces, the oceans, and , by extension, Earth's entire eco-system, would die quickly.
When Commander Koenig flies out to check the problems in Disposal Area One, the activity there lights up the sky in bright flashes from beyond the horizon. However, there is no atmosphere on the Moon, and therefore nothing to be "lit up". There would be no visible effect beyond line-of-sight.
In the waste disposal sites were on the "far/dark side of the moon", the explosion would have pushed the moon towards the earth instead of directly away as the far side is the side that is always facing away from earth.