Did Saruman Believe He Could Master The One Ring?

[Raynor]

I was rather of the impression that the conversation was not one that others present would necessarily be able to hear.

You may be right that Saruman can't listen to their mind to mind conversation over the palantir, however, we should note that in the text it is said: "his lips moved soundlessly for a while". I think it is reasonable to believe that poor Pippin was lip-synching to the messages he transmited to Sauron.

Given how important finding the Ring was to Sauron, don't you think that, if he did not believe this Hobbit to be the Ringbearer, he would have asked straight away about the Ring?

This may be explained by the above issue.

Under pressure from one of the most powerful beings in Middle-earth, Pippin could have blurted out everything he knew to make the eye and the voice go away. He did not.

As I argued previously, what he did or did not say was stricly in relation to what Sauron asked. I opine that his free will was suspended in that conversation.

Edit:
Cross-posted with Boromir88:
You raise an interesting point: indeed, it would be foolish for Saruman to show the hobbit just to tease, if he didn't already have the ring. Then again, it may be that Gandalf left out the possibility of Saruman "trading" the hobbit, even though without the ring, only with information.