Question

The claim of papal supremacy held that...

Answer 1

By reason of the Pope's office as Vicar of Christ and as pastor of the Christian Church, has full, supreme power over the Church. This is a power that he can always use. 

Answer 2

This is a very complex doctrine that says that the Pope's successor is a legitimate office and that the Bishops of Rome were to choose that successor. What makes it complex from a theological point of view is that Peter was (it is believed) the Bishop of Antioch and it was on him that Christ proclaimed his church. So the succession could be argued to be the Bishop of Antioch to the present should have been the Pope. All of this is based on Matthew 16:17 - 19 if you are at all interested. It is a very famous verse.

Not only that there is another problem. It took 150 years for the Papacy to be established and some time after that to declare how it should be done.