“When did the RCC begin to claim that Peter was the first pope of Rome?”
Cyprian and 80+ overseers that met with him at the Council of Carthage discussed Stephen’s claim to be the bishop over all bishops. They rejected his claim …
For neither does any of us set himself up as a bishop of bishops,4675 nor by tyrannical terror does any compel his colleague to the necessity of obedience; since every bishop, according to the allowance of his liberty and power, has his own proper right of judgment, and can no more be judged by another than he himself can judge another. (Ante-Nicene Fathers, Vol. V)
In a letter to Cyprian, a bishop Firmilian of Caesarea in Cappadocia, says …
And in this respect I am justly indignant at this so open and manifest folly of Stephen, that he who so boasts of the place of his episcopate, and contends that he holds the succession from Peter … (Cyprian Epistle LXXIV)
So at least by AD 250 or so Stephen claimed to hold a succession from Peter. That claim was rejected even then.
Note 1: The reason that Carthage held a council to deal with Stephen’s claims is because Rome was the nearest apostolic church to Carthage. Carthage was in Stephen of Rome’s jurisdiction. Firmilian, writing to Carthage from Caesarea, was much less respectful concerning Stephen than Cyprian was. That may have been merely personality, but I suspect it has more to do with Firmilian not considering Stephen his “superior” in position.
Note 2: By 250 it is possible that even eastern bishops would have given honor to the bishop of Rome as “first among equals.” I doubt it’s true, but it’s possible. By the fourth century, it was certainly so, but by then Rome’s authority was based more on the fact it was the capitol of the empire than on any apostolic tradition they still held.
Is Peter the Head of the Church?
Then you asked, “How about the claim that Jesus made Peter the head of the church?”
Solid claim except for the terminology, in my opinion. Jesus in the head of the church, not Peter. The early church did, however, consider Peter to be the first among the apostles, and they did reference Matthew 16 in saying that.
The problem is tying that leadership to Rome. Yes, Peter was in Rome. However, Cyprian’s Treatise on Unity states that he believes Peter’s leadership passed to all the bishops together. Orthodox churches would say that it is Antioch that received a succession direct from Peter.
Anyone who reads the early Christian writings without bias is going to see that the issue was the apostles. They were the lone authorities for the church, and the apostolic churches—not just one apostolic church—carried their tradition and were to be consulted. The original plan was for the apostles to transmit their teaching to the churches by means of the overseer and elders (or just elders in some cases) and these church leaders would pass it on (unchanged because you cannot improve on the apostles) to their successors.
It worked pretty well for a while, but by the fourth century, emperors and politics were so much a part of the succession of bishops that any handing down of truth cannot be trusted. A little historical and biblical research verifies this. The Catholic and the Orthodox both have statements—the RCC in Vatican II—saying that the faith is to be preserved unchanged, but both also have statements explaining why they are allowed to change it.