Could the Moon sit on the Earth? Let say that the Earth's Moon was knocked out of orbit and landed on the Earth with some damage but not world ending. Would Earth's atmosphere envelop to include the Moon? Would the moon sink into the Earth? What would happen?
It can't happen. The moon is quite big and can't get knocked out of orbit easily.

If it were somehow knocked out of orbit and collided with the Earth, the energy released would melt the entire crust. Huge amounts of matter would be knocked into space. After everything had settled down, you would get a lifeless world, still spherical but now slightly larger.

The moon can't get knocked out of orbit.  If it did, It can't "land". If it did "land" it would be crushed by its own weight, and all life would end in one way or another.