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?
This is a highly improbable scenario, since the Moon has a lot of momentum. However, let's say you teleport the Moon onto the surface of the Earth. Then Moon can't just sit on the Earth, and it will collapse under its own weight, probably killing everyone on Earth. The two would probably merge into a spherical shape (as most massive objects do).
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.
At the scale of the Earth or the Moon there are no solid objects. The Earth does not behave like a solid, nor does the Moon.
You have to think of these objects as more like piles of sand, made round by their own gravity. So if one round pile of sand touches another round pile of sand, they just merge and make a bigger pile of sand, also round.
the Earth's Moon was knocked out of orbit and landed on the Earth with some damage but not world ending
That's essentially impossible. I can't think of any mechanism that would allow the Moon to gently touch the Earth. Regardless, both the Earth and the Moon would be severely bent out of shape before they even begin to touch - they would look more like eggs, or a teardrop shape, then they merge, and that's the end of everyone and everything on this planet.