If  I own property in the following 4 countries , how far up and how far down 
do I own/ have control over?

 Countries
USA
Canada
Republic of Ireland
UK
Cheers

Mongolia

It would depend upon the region, state, or even town, in addition to
what the zoning for the property is (commerical, residential, etc.).
Listing all the laws for property owners isn't really practical...Is
there something more specific we can give you?

Kriswrite

mongolia --

As kriswrite-ga correctly notes, your question is much more complex
than it appears on first reading.  In the U.S., there are many lawyers
who have put their children through college on the fees they have made
opining on, and negotiating about, "air rights" and "mineral rights."

Most private real property in the U.S. is owned "in fee," which means
that "air rights" and "mineral rights" are owned with the surface
property and can be sold or leased separately (and often are). 
However, the extent of air rights has been limited by the U.S. Supreme
Court to accommodate air travel without negotiating with every
homeowner on the route.  Also, the government typically reserved a
hodgepodge of mineral rights when it originally "granted" land to
homesteaders in the West, so that some private landowners today in
that region do have complete mineral rights to their property.

Sorting all this out for the U.S. in a coherent and useful way would
be hard enough, and doing so for the other countries on your list
would compound the complexity.

markj-ga