[The Squatter of Amon Rûdh]

Gandalf says himself why he did not use the Palantír:

I have been saved by this hobbit from a grave blunder. I had considered whether or not to probe this Stone myself to find its uses. Had I done so, I should have been revealed to him myself. I am not ready for such a trial, if indeed I shall ever be so. But even if I found the power to withdraw myself, it would have been disastrous for him to see me, yet - until the hour comes when secrecy will avail no longer.

He passes the stone to Aragorn because, as Elendil's heir, he is its rightful owner. Indeed Aragorn says as much:

'Dangerous indeed, but not to all,' said Aragorn. 'There is one who may claim it by right. For this is assuredly the palantír of Orthanc from the treasury of Elendil, set here by the Kings of Gondor. Now my hour draws near. I will take it.'
Gandalf looked at Aragorn, and then, to the surprise of the others, he lifted the covered Stone, and bowed as he presented it.
'Receive it, lord!' he said: 'in earnest of other things that shall be given back...'

I see no reason to argue with Gandalf: Sauron cannot know that he has come back, which is incredibly unlikely despite Gandalf's Maiarin status. It is a special dispensation of the Powers or Eru himself that allows him to return to Middle-earth and continue with his mission, and he returns absolutely in secret. If Sauron knew of the events in Moria, and I presume that he did, he would have discounted Gandalf from his calculations unless something happened that forcibly pointed out to him that his adversary was back and active.

Ownership of an object is an important theme in myth, the rightful owner often having more power over an object than any other. Here Aragorn has the right to use the Palantír, whereas Sauron as thief and usurper does not. This is enough to tip the balance in the favour of the owner, although it is still a close-run battle. Gandalf again fears his ability to defeat Sauron, as he did in Valinor when first the Istari were gathered, but this could be an example of humility. Often those who feel inadequate to a challenge are those who are best suited to succeed in Tolkien's world. Certainly if Gandalf even thought that he still had a chance of remaining hidden and therefore leaving a factor out of his enemy's calculations, he was right to be cautious. As it turned out, Aragorn's use of the Stone led almost directly to the attack on Minas Tirith, which emptied Minas Morgul of most of its garrison just as Frodo, Sam and Gollum were trying to pass it. Coincidence upon coincidence.

The index to the Silmarillion describes the palantíri as the work of Fëanor, and the corresponding passage reads: "...and other crystals he made also, wherein things far away could be seen small but clear, as with the eyes of the eagles of Manwë."

I hope that's cleared a few things up.