If someone owns a piece of land, do they own it all the way to the center of Earth?

No. I believe you only own a short way down (correct me if I'm wrong). Go read Tunnels.

XDDDD Never thought of that!!!1

No. The laws vary depending on where you live, but for example, you don't automatically get mineral or oil rights to whatever is beneath your land. Same goes for air rights.

You know... Now I want an apartment in the Earth's core... Why not!?! Electric bill would be high because of the AC... but... Lovely view... And what a way to impress a date... Like... Oh the sunset on a river is nice but... Lava... yeah... Magma... Don't see that every day... And I'll never have to worry about snow again! Yeah... so if anyone's renting bachelor pad down there for about $400/mo. let me know.

Well then they would all meet at the center. There's not enough room in the center for everybody's land, so who would know who's was who's-

What kind of sons of bitches are we. We're overthinking this XD

Don't think so. Probably just up to the places where all the pipes and tunnels are.