What is minimal altitude that aircraft (rocket?) can fly above any country without any permission? I mean, at what altitude ends country's airspace?
Actually, what you're asking are two different questions- According to ICAO ( ICAO Annex 11: Air Traffic Services), Class G airspace is uncontrolled and aircraft can fly in the without ATC permission. The definition if Class G is left to the regulatory authorities of individual countries (in US, it is where the other airspace classes are undefined).
Where does the airspace of the country end is a completely different one and as of now, there is no agreed definition. The United Nations General Assembly Committe on Peaceful Uses of Outer space is still in the process of defining what exactly an outer space is.
What is minimal altitude that aircraft (rocket?) can fly above any country without any permission?
100 km, probably
the Kármán line, at an altitude of 100 km (62 mi) above sea level, is conventionally used as the start of outer space in space treaties and for aerospace records keeping. The framework for international space law was established by the Outer Space Treaty, which was passed by the United Nations in 1967. This treaty precludes any claims of national sovereignty and permits all states to freely explore outer space.