The rub is that there's nothing particularly Constitutional about it either, which is at the heart of the debate. Imagine that the British Ambassador, Military Attache, etc. to the US has a child with his spouse that is born at a U.S. Hospital on U.S. soil. Certainly few would argue that the child should not have any issues returning to England someday as a British subject, with British Nationality conferred through his/her parents. Certainly the case could also be made for dual-citizenship, and arguably for "natural born" status here in the U.S. (i.e. "anchor baby").
This is something that our Founders did not specifically address in the Constitution, although a fair case can be made that they did address it elsewhere, or relied on generally accepted definitions of the day. The 14th Amendment muddied the waters as much as they clarified them, but with it remaining in some degree of ambiguity, there is some substance to the argument that Obama's dad, being a British subject, may have conferred British nationality on Obama simply by fathering him. Likewise, US law at the time did not automatically confer US citizenship due to his mother's age and other possible factors, depending on where he was actually born.
I've never understood this line of reasoning. Apparently, having a foreign nation declare you as a citizen bars you from the Presidency? Doesn't that open us up to every two-bit dictator in the world deciding who can be President? After all, what if Hugo Chavez declares Sarah Palin as a citizen of Venezuela - according to this logic, she would be barred from running for the Presidency or Vice Presidency.
How another nation considers a person should be irrelevant; the only thing that should matter is how WE consider that person. From what I can tell, if you're born on US soil (or the equivalent, like a military base or possession) to at least one US citizen (naturalized parent or not) you're a natural born citizen.
Another nation's "claim" on your citizenship is irrelevant.
“there is some substance to the argument that Obama’s dad, being a British subject, may have conferred British nationality on Obama simply by fathering him”
But there is no substance, or ver, very little, to the idea that dual citizenship bars one from the office of the presidency. It in no way bars you from being a citizen at birth, after all, since whether other countries can claim you is of no concern to the U.S. government. For it does not recognize dual citizenship.
No it could not, since the people you mentioned enjoy diplomatic immunity and are not subject to the jurisdiction of the United States. Per U.S. law and the 14th Amendment they are not citizens at all.