[Morwen]

On the question of the One's existence after Sauron's defeat at the end of the second Age, I don't know that Sauron ever thought the Ring was actually destroyed. He must have been aware that in allowing much of his power to pass into the Ring that he was making himself vulnerable. Should the Ring ever be destroyed, he would be destroyed as well. He certainly seems aware of this when Frodo claims the Ring in Mt. Doom. I don't have my LotR with me but there is a passage that describes how at that point Sauron abandoned all thought of his war and his calculations and his mind became focused on his Ring and the potential peril that he was in. So I agree with the view that his continued existence in Middle Earth should have been proof enough that his ring was still around. Lost, perhaps, but still around.
The extreme difficulty in destroying the Ring should also have informed his thinking. One of the reasons Sauron may have been willing to risk putting a great deal of his power into an object is because he was fairly certain that there was only the tiniest chance that the object could in fact be destroyed. The Ring could only be unmade at one specific place - the fires where it was originally forged and then only if someone possessed the will power to toss it in there. I think Sauron would have considered the latter possibility too remote to be entertained.