What exactly is Peter's connection to Rome according to Catholic dogma?

Was Peter ever in Rome and did he die there?

There is no doubt that Matthew's Gospel tells us that Jesus nominated Peter as the rock on which he would build his church. So, whatever city could claim Peter as its patron would have a huge advantage over other cities in the Christian world. Tradition has credited various of the apostles with remaining a a number of important Christian cities, sometimes two or three at the same time, and leading the church there. One tradition is that Peter became the bishop of Antioch, but in the end Rome, the eternal city, became established as the apostolic city of both Peter and Paul.

Yet another tradition is found in The Apostolic Constitutions (VII.xlvi), which purports to provide evidence that Peter was in Rome and appointed Clemens as successor to Linus, who was actually appointed by Paul:

Of the church of Rome, Linus the son of Claudia was the first [bishop], ordained by Paul; and Clemens, after Linus' death, the second, ordained by me Peter.

The Liber Pontificalis contains purported biographies of all the early popes, listing Peter in first place, but in no sense was this ever held to be an official document of the Church. It says that Peter went to Rome when Nero was emperor (54-68) and occupied the chair for 25 years, 2 months and 3 days. On this evidence, Peter was indeed bishop of Rome, although he was not executed by Nero.

Decrees of the First Vatican Council (1868), clearly defensive in nature, talks at length about the primacy of St Peter, bestowed on him by Jesus, of the pope as successor to Peter and of the "see of Saint Peter," an implied reference to Peter as the first bishop of Rome. Session 4, chapter 2 says of the Holy See, "which he founded and consecrated with his blood," so the bishops meeting in synod did not need to state that Peter had been in Rome, although they believed this to have been the case.

The Catechism of the Catholic Church gives a good overview of the Catholic position, at clause 936:

936 The Lord made St. Peter the visible foundation of his Church. He entrusted the keys of the Church to him. The bishop of the Church of Rome, successor to St. Peter, is "head of the college of bishops, the Vicar of Christ and Pastor of the universal Church on earth"

From 877 to 892, the Catechism frequently refers to the bishop of Rome as Peter's successor, but once again does not directly claim that Peter was ever in Rome.

The Catholic Encyclopedia states:

It is an indisputably established historical fact that St. Peter laboured in Rome during the last portion of his life, and there ended his earthly course by martyrdom. As to the duration of his Apostolic activity in the Roman capital, the continuity or otherwise of his residence there, the details and success of his labours, and the chronology of his arrival and death, all these questions are uncertain, and can be solved only on hypotheses more or less well-founded. The essential fact is that Peter died at Rome: this constitutes the historical foundation of the claim of the Bishops of Rome to the Apostolic Primacy of Peter.

The Encyclopedia goes on to briefly explain the existence of second-century traditions that Peter had gone to Rome. There is undoubtedly a second-century tradition that Peter went to Rome and died there, but at such a great distance from the apostolic era, this is not indisputable evidence. Moreover, there are minor contradictions between these traditions. The Encyclopedia refers to the fourth-century "Chronicle" of Eusebius as stating that Peter and Paul died in 67 CE, but also notes the "Chronograph of 354" to say that they died much earlier, in 55 CE. The general tradition is that peter, as bishop of Rome, appointed Linus as his successor, but Irenaeus, bishop of Lyon towards the end of the second century, saw things a little differently in Against Heresies, III.3.2-3, where he wrote that Peter and Paul jointly appointed Linus. Irenaeus was trying to convince his gnostic opponents in Lyon that he represented the true legacy of Jesus, so it may be unsurprising that he claimed that Peter and Paul founded the church in Rome and that the bishop of Rome was appointed by both "glorious apostles."

It appears that the best evidence that has been put forward to prove that Peter really did go to Rome is from 1 Peter 5:13, which says that Peter was writing from "Babylon." John W. O’Malley, S.J. says, in A History of the Popes, page 8, that no one piece of evidence states in straightforward and unambiguous language either that Peter either went to Rome or that he died there and, even when relying on 1 Peter, he acknowledges this might not have been written by Peter himself, but claims it was at least written under his inspiration.

The acknowledgement that 1 Peter might not have been written by Peter himself opens a can of worms, as it introduces the scholarly view that the epistle was actually written in Peter's name long after his death. Bart D. Ehrman says, in Forged, page 68, scholars have long realised what this reference to ‘Babylon’ means. Babylon was the city that was seen as the ultimate enemy of God among Jews, as it was Babylon that had defeated Judah and destroyed Jerusalem and its Temple in the sixth century BCE. By the end of the first century Christians and Jews had started using the word Babylon as a code word for Rome, the city that was the enemy of God in their own day, which also destroyed Jerusalem and its Temple in the year 70. Peter, or even a secretary, would not have called Rome "Babylon" before the Romans had destroyed Jerusalem in the year 70, so on this ground alone 1 Peter is pseudepigraphical and can not be relied on as evidence for Peter in Rome.

A site under St. Peter's Basilica includes several graves and a structure said by Vatican authorities to have been built to memorialise the location of St. Peter's grave. If authentic, this would prove beyond doubt that Peter had gone to Rome and been buried there. Between 320 to 327, Constantine built a basilica over the early Christian necropolis that was purported to be Peter's resting place. In 1942, the Administrator of St. Peter's, Monsignor Ludwig Kaas, found previously overlooked remains in the monument and, having little understanding of correct archeological procedures, secretly ordered the remains stored elsewhere for safe-keeping. In doing so, he destroyed any certainty that the bones actually originated in the tomb as claimed. Based on bone tests that showed the remains belonged to a man in his sixties, Pope Paul VI announced on June 26, 1968 that the relics of St. Peter had been discovered. As Pope Paul was not speaking ex cathedra, his statement is not regarded as infallible.

Was Peter ever a bishop?

As mentioned previously, we have a tradition that Peter was bishop of Antioch, a role that would haver required him to remain in the city for some years, and we also have tradition that Peter was bishop of Rome. The first tradition is antithetic to the Roman Catholic position and receives little support, at least in the West.

Francis A. Sullivan S.J. says, in From Apostles to Bishops, page 15, that most Catholic scholars say that the church of Rome was most probably led by a college of presbyters until well into the second century, with no evidence of a ruling bishop in the first century. He does not speak for the Church as a whole, but in that statement does identify the position of prominent Catholic scholars. I have carefully read the arguments in his book and find them convincing. Father Sullivan does not contradict Church doctrine that Peter was the first bishop of Rome, but it seems inconceivable that Peter would lead the church in Rome, as its first bishop, and appoint a successor as bishop, only for the Christian community of Rome to ignore this precedent for up to a century after the death of Linus.