I think it depends on context. Because we US-residents seem to love referring to oursleves as Americans, as opposed to Canada-residents being just "Canadians."
Nope. American generally refers to America (United States of). Everywhere else is referred to by its country name. This is because the US part is usually dropped because most countries preceed their name with discriptors or honorifics ( Federation of, Sovereign state of, etc).
If you go to most South American countries, the people refer to themselves as, "American." I'm not sure about Canada or Mexico, but most other american countries have, "American," citizens.
... I don't believe so. The names I hear thrown around are national like Chileano or Mexicano, with "Americano" reserved for us folks from "Los Estados Unidos"
If so, then it must be continental, like folks from Germany calling themselves European. More people, I think, would identify with their country than their continent (unless they live in Australia, in which there isn't much difference).