USAF definition--The US Air Force defines the atmosphere-space boundary at 81 km.
FAI definition--The Federation Aeronautique Internationale defines the atmosphere-space boundary at 100 km. As of the end of 2004, 437 people have been this high or higher (according to Encyclopedia Astronautica). Several governments use an altitude at or near this in regulating space activities.
When you can orbit the Earth--where atmospheric drag is low enough to allow a satellite to complete an orbit; this depends on the characteristics of the satellite and the condition of the atmosphere. About 120 to 150 km is typical.
When air molecules are not gravitational bound to the Earth--this is the beginning of the exosphere and is 300 to 600 km high, depending on solar activity.
Where you say it does--in 1976 the nations of Columbia, Equador, Brazil, Congo, Zaire, Kenya, Uganda, and Indonesia claimed sovereignty to an altitude of 35,862 km. The UN rejected these claims in 1980.