: 1) Where did his pick up his knowledge of ring making?

Potential sources of knowledge would have included:

1. Any surviving Ring-smiths in Lindon, Imladris, and Lorien
   
2. Any surviving books on Ring-lore and the history of the Gwaith-i-Mirdain in Imladris, Lorien, and Lindon
   
3. Any surviving books in Gondor
   
4. Possibly the Palantir of Orthanc
   
5. Saruman's own knowledge gained as a Maia in the following of Aule
   

: 2) Was this ring subject to the power of the One Ring?

If it was made according to the lore of the Rings of Power, it would have to be.

: 3) Did Saruman's ring survive the destruction of the One Ring?

Do you mean physically? It should have. But its power, if made according to the lore of the other Rings, should have failed. He does seem to imply that his Ring failed in "Many Partings", although his words could easily refer only to the military power he had amassed -- but then, Gandalf, Galadriel, and Elrond were not really involved in the Ents' war with Saruman:

  "'For a moment his eyes kindled.  'Go!' he said.  'I did not
  spend long study on these matters for naught.  You have doomed
  yourselves, and you know it.  And it will afford me some
  comfort as I wander to think that you pulled down your own
  house when you destroyed mine.  And now, what ship will bear
  you back across so wide a sea?' he mocked.  'It will be a grey
  ship, and full of ghosts.'  He laughed, but his voice was
  cracked and hideous.

: 4) Was this ring taken off him by Treebeard when Saruman left
: Orthanc?

You're guess is as good as anyone's, I think. But I don't believe Treebeard knew about the Ring.

: 5) How powerful was Saruman's ring?

That is a matter for speculation. However, it seems not to have been as powerful as Sauron's, for otherwise he would not have wanted the One Ring (perhaps -- maybe he would have anyway).

: As far as the power of Saruman's ring goes, who knows? It was
: obviously not too powerful. Gandalf overwhelmed the power of
: Saruman in their confrontation in The Two Towers and this must
: have been a Saruman with both his staff AND his ring. But then
: I would believe that one of Gandalf's specfic tasks when he was
: sent back would have been to deal with Saruman and therefore
: gandalf was enhanced to the level which would mean he
: could take out Saruman, despite his staff and ring.

Yes -- Tolkien stated specifically that Gandalf was sent back with increased power and authority. He was more powerful as Gandalf the White than he had been as Gandalf the Grey.

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Parma Endorion: Essays On Middle-earth, Revised Edition