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According to International Law, to what altitude does the jurisdiction of a state extend? Is there a definite limit mandated either through treaties or court judgements?
After reaching that limit, do we find ourselves in the outer space regulated by space law? Is there a layer of international space, regulated by other treaties?
This is a very practical question: To what altitude may launched spacecraft or satellite ascend/descend without violating the air territory of a sovereign state?
The general rule of thumb on this type of issue is that if it is in orbit, then you don't have to worry about traveling through other countries, and if it is not yet in orbit, you should ask permission from whatever country you are flying through. For this reason, and safety reasons in general, most spacecraft launch from the coast. Thus, a sub-orbital flight is still maintained by the country below it, but an orbital flight would not be.
Of course, like ships, Orbiting vehicles are subject to their registered nation, but that's another issue entirely.
A few other things of some interest:
There is no internationally agreed-upon definition of "outer space", or delimitation between outer space and airspace. The 1967 Outer Space Treaty does not contain a definition of the term "outer space". This was intentional, because states have significantly greater freedom of action in outer space than they do in airspace. Specifically, the Outer Space Treaty provides for freedom of use and access in outer space, while airspace is sovereign territory. Bottom line is that you can overfly another country if you do it in outer space, but not if you do it in airspace. For that reason, some countries have intentionally prevented the formation of an internationally agreed definition of outer space in order to maintain freedom of action in very low but still basically orbiting activities.
In practice, however, these edge cases aren't at all common, and the lack of a delimitation between outer space and airspace hasn't caused any practical problems. That's another reason why some countries don't see a need to answer the question.
There are countries that have adopted a definition for their own legal purposes, however. These definitions generally follow either an altitude-based approach (generally tracking the Karman line) or a functionalist approach, under which outer space is considered the altitude at which orbital flight is possible without utilization of atmospheric lift, or something to that effect. But these definitions all vary and there is no international standard.
With regard to objects in GEO, the states (here I mean countries) located below them do NOT have some kind of legal claim or jurisdiction. Some countries tried to argue this a while back, and those arguments have all been soundly rejected. Communications satellites in GEO are subject to frequency and orbital allocation rules set out by the International Telecommunications Union, as well as being regulated by their launching state, unless operations have transferred to an entity located in another state. Otherwise, the general rules about freedom of use and access set out in the Outer Space Treaty apply.
So, to simplify: if you go straight up and straight back down (more or less), no one is going to care other than the country you launch from. If you go above the karman line, some countries will think you went into outer space, but hey, you have freedom of use and access there anyway. If you want to overfly a country, you need to either be unambiguously in outer space, like high enough to orbit a bit without propulsion or lift effects, or at an altitude that the country in question itself considers to be outer space.
