When the Vatican sends representatives to other countries, are they representing the political arm of the Church or the religious arm? THe same thing with treaties, concords, etc.?
Ambassadors and treaties help to protect the rights of the Church and its members in the various countries involved. Not really much different from the Mormon church, say, or the Baptist church, sending lobbyists to Washington to protect the interests and freedoms of those religious organizations and their members. It just so happens that in this case, the head of the religious organization involved is an independent sovereignty. Back in the 19th Century and before, when the Papal states were extensive and had true subject populations, your Islamic analogy would be more persuasive, but a few acres in the city of Rome is hardly the same thing.
You've actually hit on something that causes a great deal of annoyance within the Church. I think it was Paul VI who elevated the Secretariat of State over the Congregation of the Doctrine of the Faith as the highest ranking curial department.
We've had difficulty focusing ever since.