It had nothing to do with St. Peter, there being no developed ancient tradition that he ever set foot in Egypt (the Coptic legend of today seems to be connecting 1 Pet. 5:13 with Babylon in Egypt (Old Cairo). And he certainly was not there before he was in Antioch, which is recorded in Holy Scripture. Alexandria, unlike Antioch and Rome, has never claimed to have been founded directly by St. Peter.
I'll provide a quote at the end for this. Alexandria inherited the "faith of St Peter" from the evangelist St Mark, who was St Peter's disciple. A line of Apostolic succession, though not physical presence.
The succession of St. Peter at Alexandria never claimed that St. Peter was ever there, just that his disciple St. Mark founded the Church of Alexandria. Odd thing that the traditions of those three sees never speak of Jerusalem, where St. Peter obviously was and received his consecration. Alexandria never precedes Antioch in the history of the Apostles, but it did in the imperial ordering of the metropolises.
You're correct. St Mark, St Peter's disciple, brought "St Peter's faith" to Alexandria. This is why is is recognized as such.
He was succeeded by St. Linus, who was consecrated by St. Paul.
Debatable due to resources.
The Apostolic Constitutions says that Linus was the first bishop of Rome and was ordained by Paul, and that he was succeeded by Clement, who was ordained by Peter. Cletus is given as Linus's successor by Irenaeus and others (St Jerome, Eusebius, John Chrysostom, the Liber Pontificalis, etc) who present Linus either as the first bishop of Rome or, if they give Peter as the first, as the second.
As for having the presence of St. Peter to consecrate bishops until his martyrdom, that doesn't say anything after his martyrdom. Even according to the Vatican's own rules, a pope doesn't have to become pope in Rome (otherwise the Avignon papacy blows a nearly century hole in the institution),
Rome has never had anything to do with the Papacy, other that that being the current seat. The papacy isn't Rome, the papacy is in Rome. If the Patriarch of Moscow became Pope, he could sit in Russia all day long, and still be the Pope.
he becomes pope immediately when he accepts his election (which he doesn't have to do in Rome, nor does he have to be elected at Rome). This is not like the Aaronic High Priest, who had to be consecrated in Jerusalem, nor the succession of the elders of Israel, who had to lay hands on their successors in the Promised Land. Hence the basis of the relevance of where St. Peter was martyred does not exist to support the claims of the papacy.
It wasn't mentioned to support the papacy. It was mentioned to support that he was ever in Rome. That was doubted by the previous poster.
Whenever apologists for the Vatican try to gloss over the fact that St. Peter founded the See of Antioch and through his disciple St. Mark founded the See of Alexandria, the last refuge is that St. Peter was martyred at Rome. They never explain, however, how, according to what the Vatican teaches about its papacy, that matters.
That the manner that the Vatican has selected the holder of the papacy as it defines it has changed over time creates difficulties in examinging this question. Yet we can start with the present canons on the matter:
Can. 331 The bishop of the Roman Church, in whom continues the office given by the Lord uniquely to Peter, the first of the Apostles, and to be transmitted to his successors, is the head of the college of bishops, the Vicar of Christ, and the pastor of the universal Church on earth. By virtue of his office he possesses supreme, full, immediate, and universal ordinary power in the Church, which he is always able to exercise freely.
Can. 332 §1. The Roman Pontiff obtains full and supreme power in the Church by his acceptance of legitimate election together with episcopal consecration. Therefore, a person elected to the supreme pontificate who is marked with episcopal character obtains this power from the moment of acceptance. If the person elected lacks episcopal character, however, he is to be ordained a bishop immediately.
Can. 333 §1. By virtue of his office, the Roman Pontiff not only possesses power offer the universal Church but also obtains the primacy of ordinary power offer all particular churches and groups of them. Moreover, this primacy strengthens and protects the proper, ordinary, and immediate power which bishops possess in the particular churches entrusted to their care.
§2. In fulfilling the office of supreme pastor of the Church, the Roman Pontiff is always joined in communion with the other bishops and with the universal Church. He nevertheless has the right, according to the needs of the Church, to determine the manner, whether personal or collegial, of exercising this office.
St. Ignatius of Antioch, tradition tells us, was consecrated by St. Peter at Antioch. Tradition also would affirm that St. Peter was always joined in communon with St. Ignatius. St. Ignatius of course had "an episcopal character"-which isn'tt needed anyways-as did SS. Linus and Clement in Rome. St. James in Jerusalem was also still alive at Jerusalem, where he had presided over the Council of Jerusalem and rendered its decision. Tradition tells us the the Apostles SS. Peter, James and John consecrated St. James to his position as bishop of Jerusalem. So even if we are focused on St. Peter, there are plenty of candidates: there is no reason why St. Ignatius or even St. James could not be successor to St. Peter as "pope": after all, Pope John Paul II was Archbishop of Krakow when he was elected (and tradition identifies St. Paul as the consecrator of St. Linus, his companion 2 Timothy 4:21). Why not take St. Mark as St. Peter's successor? Tradition, even quoted by Pope St. Gregory, calls St. Mark the disciple of St. Peter and indeed considers Alexandria-where the title/office of the papacy originated-a Petrine see because St. Mark founded it at St. Peter's instruction, if the supremacy supposedly went from Jerusalem to Antioch to Rome with St. Peter, why not to Alexandria with his discple, the Evangelist St. Mark? Supposedly the soujourn in Avignon didn't affect the papacy, so why not move to Alexandria?
The teaching of the Vatican depends on a division between the papacy and episcopate based from St. Peter. Otherwise, there is no reason why the Patriarch of Antioch isn't THE successor of St. Peter. The Vatican gets around this by combing the papacy as the successor of St. Peter with the episcopate of the bishop of Rome. But we have no basis in the traditions-laying aside for a moment the problem that there was no papacy at that date-to make such a combination.
So, what is the basis for claiming primacy for the successor of St. Peter at Rome and not for the successor of St. Peter at Antioch, when, according to Vatican dogma, there is no requirement to be in Rome?
Btw, interesting that at the same time the issue came up at the highest level of the Empire: St. Peter was martyred during the persecusion of Nero. When Nero committed suicide, in the Year of Four Emperors, Emperors for the first time took office outside of Rome.