Our puppy does something very strange that I think is a way of showing us that he is below us in the pack order, but I really don't see him ever doing such a thing in a real pack of dogs:

If any of my family lies on the floor face up, he will come over to us and lay his neck over our mouths. Is this submissive? Baring his most vulnerable point to our mouths? Or is it something else?

Usually putting its head above another dog's head is an act of dominance. I hope you keep him well enough brushed you don't get a mouthful of hair.

Yes the hair isn't a problem, He only ever does it when we're face up. It's always over our mouths.

Cappy, labman is right. His behavior is a dominant one, not submissive. Don't let him do that. Correct him immediately by turning him over on his back and holding him in that position, with you above him. As far as the other dogs, he recognizes that he is not dominant with them, therefore, he doesn't do it.