That's been a longstanding rumour, but he's on record as denying having talked about it.

Doesn't count.

He didn't need to be awarded it by the pope, as he simply assumed it on the death of his father in 1766. He had no legitimate children and was succeeded as Jacobite claimant by his brother Henry 'IX' and then by his remote cousin Charles Emmanuel IV of Sardinia, counted by Jacobites as Charles 'IV' though this was not a claim he pressed.

Yes. The papacy was far less interested in being seen to support his claim than it was that of his father.

Neil Oliver on the BBC series History of Scotlan (the relevant episode of which I just watched) said that the Pope recognized BPC's father as James III & VIII of England, Scotland and Ireland; upon James' death in 1766, Charles asked to be recognized as Charles III, but this was refused; however, upon Charles' death, the Pope relented and posthumously recognized his claim.

Of course, it is rather an important principle of the UK's unwritten constitution that the Pope doesn't get a say in such matters.