It does, but as more air drops down, the pressure increases.
This pressure then starts to push back against the air above it.
Which is why we have atmospheric pressure at the surface, but that goes down to pretty much 0 in space.
Even in low earth orbit there are still some particles, which causes satellites and such to slow down, requiring them to fire some thrusters every once in a while.
Yes. Space includes planets, so the gravity wells of the planets, create different densities.
You said space, and not a perfect vacuum. So a vacuum in theory has no density. But there can be some elements so far apart they act as they effectively have no density.