We're all Maiar, those tiny things you call eagles nowadays just don't remember it. At least they don't have to feed on prey that would be missed by farmers anymore.
I thought all Great Eagles were Maiar, like Balrogs or the Istari.
Maybe? It seems like a strong possibility for Thorondor and the Great Eagles of his day at least. I'm not nearly as sure about their later descendants.
Maybe Thorondor, but I doubt Gwaihir was Maiar.