Significance of the green stone?

[Hot, crispy nice hobbit]

'It needs more to make a king than a piece of elvish glass, or a rabble such as this...'

Why had Sauron assumed that Aragorn's emerald stone is of Elven make? Was he even there when Earendil began his voyage? Granted that Sauron had been hanging around the Elves during the making of the Three Rings, did he really noticed the stone?

Above all, why should it end up in Middle Earth again? I can't remember clearly where the stone came from, but I don't suppose the stone could just float back to Middle Earth again. Why should Sauron even try to make a connection between the green stone and the heir of Isildur?

I suppose Feanor or Celebrimbor made it, and then just left it lying around until Galadriel picked it up? Its role seemed extraordinarily small throughout the epic, from its discovery at the Last bridge to the crowning of Aragorn... Just goes to show that some men like jewels too.