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.
Quote from St Gregory:
"Wherefore though there are many apostles, yet with regard to the principality itself the See of the Prince of the apostles alone has grown strong in authority, which in three places is the See of one. For he himself exalted the See in which he deigned even to rest and end the present life [Rome]. He himself adorned the See to which he sent his disciple as evangelist [Alexandria]. He himself established the See in which, though he was to leave it, he sat for seven years [Antioch]. Since then it is the See of one, and one See, over which by Divine authority three bishops now preside, whatever good I hear of you, this I impute to myself.”
(Book VII, Epistle XL of St Gregory I, Pope of Rome to Pope Eulogius of Alexandria)
It was said that the three original Patriarchates of Rome, Alexandria, and Antioch were given primacy in their respective regions partly due to their Petrine origins.
Additionally, the Arabic Hudoyo Canons gives the Patriarch of Patriarchs role to the Pope of Rome. (since we seem to be arguing papacy AND St Peter in Rome)