I don't think so, if you lived near an airport how would you anchor a sign to say no trespassing!!
It depends. Look at the deed. Does it have mineral rights, for example? And I'm sure it is different in some countries.
Not in the US! You own everything up to 100 feet; anything beyond that is public air space.
Nope. There's only one country in which the land owner owns the rights to the sky and ground all the way to the core. I can't remember which one, though. Singapore, maybe. I think it was in the general area.