Re: You don't own the air above your ground
They also own land rights to the ground underneath them which is why miners have to buy or lease mineral rights from the land owners (just ask them).
Maybe. Try buying real estate in New Mexico, for example - the deed specifically excludes rights to a range of minerals. And water ownership, both surface and subsurface, is very complicated.
It's certainly not anywhere near as simple as "In the United States, ... property owners ... own land rights to the ground underneath them".