With regard to your third example, the legislation is quite clear and there is no requirement to have an EASA licnence in the circumstances you describe as no part of the flight would take place within the EU.
The legislation is of course
anything but clear. Part FCL allegedly applies to operations of aircraft
. Neither the basic regulation nor part FCL restrict the geographical scope of the operation. Even Article 4.1(d) of the Basic Regulation, which is excluded from the scope of part FCL, refers to operations inside and outside the community. And that in spite of having studied at reasonably good university in the south of France and earning a jolly tax free 10k a month in Cologne. ;-)
I would call the 3rd example not applicable because the FBO where you rent the plane would have the ultimate say over where the airplane is going and where not. As a result, the US based FBO would be the operator and part FCL wouldn't apply as stipulated in article 1 of part FCL.
It is clear from the language in part FCL that "operator" is very distinct from "pilot". For that reason, a London based Delta pilot does not require any sort of EASA certification when flying a company aircraft.
The statement
Once outside of the terriotry of EU member States the EU licensing requirements have no legal force and pilots with need to comply with requirements or the state of registry.
is bolleaux.
Mike Echo : Ignorance of the Law is never a valid defence. That being said, there are two major issues with the enforcement of this particular item of Part FCL
1) The establishment of the operator is a non-trivial legal matter.
2) There has been some legal preparation done by a number of (mostly private) operators who would vigorously fight enforcement on the basis of a number of issues, one of them being discrimination. It would be career risking for a local CAA official to start a witch hunt without rock solid political backing from their transport minister.
et ceterum censeo easam delendam.