NATO is authorized under the UN Charter (it's considered a "regional arrangement" under Art. 51 of the Charter), but it was established by its own treaties.
Do any of our military under any circumstances you outlined above wear any other insigna
Good question.
It was concieved by the un and gets it's marching orders from the un so yes, NATO works for the un.
Question...does an American president go to or get permission or at least discuss with the un possible missions before that mission is launched?