I'm writing it as I go; it is based somewhat on previous attempts by me to put this stuff together, but this time I seem to be more in the flow. I'm about to set out on a road trip so it may be a while before I can finish the "Alexandria" section, with a discussion of the actual patriarchs of Alexandria (this subsection discusses everyone but!)
We hear almost nothing about orthodox Christians at all in Egypt until the 3rd century. For a long time, the orthodox were just one, and not a very prominent one, of a bewildering variety of “Christianities” found there. Most of the sects are just names to us, and probably never had much of a membership; and from Nag Hammadi and Oxyrhynchus we have some intriguing texts which we cannot identify with any known sect, indicating that the longish lists of heretical sects we get from hostile authors are by no means comprehensive. Egypt always had an eclectic attitude toward religion: an account of a state visit by Hadrian (fresh from destroying Judea) mentions people who would attend Christian preachers, and also donate to Pythagorean monks, and also visit the temple of Serapis; the popular deity Serapis was a fusion of Osiris, one of few Egyptian gods depicted with a human rather than an animal head, with the Apis bull, the most prominent of the sacred animals viewed as divine incarnations, originally the incarnation of Ra, but later identified with Horus, falcon-headed and supposed to be the son of Osiris, but inconsistencies seldom bothered the Egyptian mind. We might compare the modern Japanese who have Christian baptisms for their children but Shinto weddings and Buddhist funerals, letting each religion handle what it does best. One somewhat-sizable eclectic denomination was the Naassenes, who believed in Judaism, Greek philosophy, and Egyptian paganism simultaneously, and claimed that the mission of Jesus was to reveal that these all taught the same truth. These, or someone similar to them, produced early versions of the "Hermetica", a literature which circulated underground in medieval times (and had not been thought until recently to go back to ancient times) in which the Greek god Hermes is identified with the Egyptian god Thoth and with Mousaias, who is both Moses and Homer's Muse, and therefore hailed by the title Trismegistos “triple great”, and reveals a farrago of alchemy and astrology. The “Treatise of the Great Seth” (supposedly revealed by Jesus) similarly identifies the third son of Adam from Genesis with Set, the Egyptian god of chaos. Some appear to have gone further with Marcion's anti-Semitic Gnosticism, reversing the roles of the good guys and bad guys in the Old Testament, like the Cainites (who thought Abel had it coming?), or the Ophites (who worshipped the snake? We only know the names). Some sects did manage to survive their founding gurus and become somewhat larger movements: we have encountered the radical Carpocratians, who outlived Carpocrates even though Epiphanes, the son whom Carpocrates groomed as a successor, died at 17, though not before penning a memorable anticipation of Proudhon's “Property is theft”: “When men forget that the world is for the benefit of all, on that day the thief is born.”
The most successful was Valentinus, who taught a complex multi-layered cosmology (this was typical of pre-Christian as well as post-Christian “Gnostic” sects in which the gnosis “knowledge” that was taught, generally for a price, consisted of the secret names, regarded as having magic powers, for heavenly and hellish realms and their presiding deities or demons) formed by emanations from “The Ineffable”; contrary to Marcion's system, the evil demiurge was not the highest emanation (that, instead, was the “First Mystery” identified with Jesus) but the lowest, called “Adam”, who was not just the first man but still exists as the ruler of hell (in some Celtic and Hindu mythologies, likewise, the first man, being the first to die, now is the king of the dead). Valentinus claimed to have derived his system from the secret teachings of Paul, as passed down through Theudas. Theudas called himself the re-incarnation of the prophet Elisha and the successor to John the Baptist, who was Elijah; he had the Baptist's robe, as Elisha had inherited Elijah's, and promised in the late 40's AD that he would part the Jordan river with it, enabling the Baptist sect to cross over and retake the Promised Land from the Romans, but no miracle occurred and Theudas was executed along with many of his followers. Like the Zealots, Theudas would seem to be a competitor to the Christians rather than an ally, yet it seems that everybody knew everybody else: one fragment from Nag Hammadi hails Theudas, and his wife Mary Magdalene (!), and their children (we would love to know how the rest of that text went). In the major Valentinian scripture, the “Questions of Mary”, Magdalene is visited for 14 years after the crucifixion (up to the time of Theudas' death?) by Christ descending from the realm of the First Mystery to teach her about all the higher heavens. Irenaeus says this came in “lesser” and “greater” versions: the text we have (commonly called Pistis Sophia “faith-wisdom” from a long parable about how the divine emanation Pistis Sophia came to be trapped in the material world) is presumably the “Greater Questions of Mary” since it is very long (no doubt the result of substantial re-working by later Valentinians). Valentinus visited Rome, where Pope Victor condemned him, but he obtained a following in northern Italy and Gaul, perhaps with support from the patriarchs of Arles (the curious story of Mary Magdalene sailing to the south of France might derive from an early attempt by Arles to re-position themselves as successors of Magdalene rather than of Paul), hence the space Irenaeus devotes to combatting them; and as noted above, disciples of his founded a long-lived movement in Asia. Their depiction of Christ as a wholly spiritual being, whose physical manifestation was really a delusory phantasm (called the “docetic” view), was influential among non-Valentinian Gnostics as well, as condemned by Polycarp's line “Who is anti-christ but the one who denies Christ came in the flesh?” also found in 1st John (the orthodox view is that Polycarp was quoting 1st John, but I think the author of 1st John was quoting Polycarp, since I doubt that any “docetic” Gnostics yet existed while John was still alive). It is curious, then, that Irenaeus says that of the gospels, the Valentinians only use John (as, he says, the Ebionites only use Matthew and the Marcionites only use Luke); perhaps this is why chapter 21 (with Jesus eating fish) was added to that gospel (emphasizing that even the risen Jesus needed physical sustenance).
Basilides may or may not have had an organized following, but his “gospel of Mathias” was widely copied enough to earn a blunt rejection, along with the Valentinian and Montanist scriptures, in the Muratorian Canon: “From Valentinus, and from Basilides, and from the Cataphrygians, we receive nothing at all.” We now have fragments indicating that it was in the form of interpreted quotations, “Jesus said... and this means...”, where the sayings of Jesus are the familiar “Q” material (which I believe is what Matthew/Mathias actually wrote) and the commentary is a Gnostic spin. Fusions of “Q” with some version of the “gospel of Mark” existed early in the 2nd century, in “Hebrew” (probably, rather, mostly Aramaic heavily salted with Hebrew words as in much religious speech of the time) among the Ebionites and in Greek among the orthodox, but the fused work was not always attributed as a whole to Matthew at first. Justin Martyr c. 150 quotes a gospel which includes the “star of Bethlehem” story (though not the “guards at the tomb” story) otherwise known only in the canonical gospel of Matthew, but calls it just “the memoirs of the apostles” without singling out a particular apostle as the author; while the Muratorian Canon apparently thought it was by Andrew. Unfortunately the Muratorian text is missing the entire description of the first gospel, and all but the last sentence “from what he remembered and what he was told” of the description of the second gospel, but in describing the “gospel of John” it gives a curious story of how John had a dream revealing that he and Andrew would be the two disciples who would write gospels. In “Acts of Barnabas”, however, Barnabas is said to have circulated the “two books of Matthew”: this is probably good information, since there would be no motive to make it up (rather the reverse; there would be good motive to conceal it), so we have to wonder what the “other” book of Matthew was. I think it was a re-editing of the gospel of Mark (without an attribution to Mark, who wasn't yet “anybody”) to fix up some grammar: Mark doesn't write “bad” Greek but sometimes uses Aramaic sentence-structures (Greek was clearly a second language for him; I do not, however, accept Lamsa's proposal that the Aramaic translation of Mark in the “Peshitta” is the original); Matthew and Luke share, beside the “Q” material, the “Minor Agreements” (common edits to the Markan narrative). Another tale has it that Barnabas was buried with his own distinctive “gospel” on his chest; a late medieval forger (a Sicilian convert to Islam) used this story to concoct a blatantly pseudepigraphic “gospel of Barnabas” which some Islamic web-sites try to pass off as genuine.
Barnabas (Aramaic bar-nabi “son of a prophet”) and Mark (Semitic name Yochanan “John”; the Latin rather than Greek alternate name Marcus probably reflects an earlier family client-relationship to Mark Antony) were important first-generation Christians, from a mercantile family operating in Cyprus and Cyrene (eastern Libya): those two places had important economic ties, and Cyprus-and-Cyrene was a single province sometimes under the Ptolemies; the demand by Cleopatra, supported by Antony, that Rome turn over Cyprus and Cyrene to Ptolemy Caesarion, her son by Julius Caesar, was the spark for the final rupture with Octavian (later Augustus Caesar; the confrontation ended with Egypt occupied by Octavian and Ptolemy Caesarion, of course, dead along with Antony and Cleopatra). Their wealth appears to have been sizable: Acts makes a special point of noting that Barnabas donated all his resources to the use of the church, and refers to a house owned by Mark's mother in Jerusalem. Barnabas is one favored candidate for the author of the epistle to the Hebrews, which was also sometimes called the epistle to the “Alexandrians” (as in the Muratorian Canon, which rejects it); it is certainly not by Paul (it is anonymous, where Paul always announces himself up-front; it is pedantic and professorial in style, where Paul is “chatty”; and it is in an Egyptian rather than Anatolian dialect of Greek) but by someone of the first generation who knew many of the same people (it is the closing salutations which caused people to think it was by Paul). But Martin Luther favored Apollos instead: Apollos was a learned Jew from Alexandria who was regarded as something of a leader (Paul deplores divisions within the Corinthian church between those following himself, followers of Cephas, and followers of Apollos). If the “epistle of Barnabas” were genuine, we could linguistically compare: but that text, although also believed to be 1st century, is not by a Jewish author; it describes the provisions of the Torah as parables, intended to teach symbolic lessons, but makes errors about the content of the Torah which a literate Jew would not. Hebrews/Alexandrians does not, unlike “Romans” as we have it, presume the existence of some Christian community there already, but addresses Jews in the hope of persuading them to accept Jesus as the Messiah (in the Jewish sense).
So, when Mark arrived in Alexandria, there may or may not have been any Christians there already, and if there were, they may have been “Christians” of a very odd sort. The oldest traditions have Mark converting late in the ministry of Jesus: Papias and the Muratorian Canon agree that his gospel was partly from what he witnessed himself, partly from what he learned second-hand. A story which the gospels of Matthew and Luke do not retain has a young man dressed only in a white robe (as in the “Secret Mark” story of Lazarus) whom the Romans tried to arrest, but he slipped out of the robe and ran away naked; if this is not Mark himself, it is hard to understand why the story was included (comic relief?). The patriarchs of Alexandria would rather have Mark convert early, as one of the servants at the wedding in Cana who poured the water into the big stone jugs and was astonished to find it turned to wine, and then be appointed one of the “seventy” disciples appointed by Jesus as missionaries in addition to the “twelve”; but this is just an attempt to heighten Mark's standing. For one thing, someone with as rich a family as Mark's would not have been working as a servant; for another, the gospel of Mark has neither the “wedding” story (only in John) nor the “seventy” (only in Luke); and the “water into wine” story is unlikely to be true, having an evident propagandist motive to show that Jesus was not simply doing magicians' tricks: Heron of Alexandria's book on magic gives seven ways of turning water into wine (evidently a staple in magicians' repertoires then), which like the “horn of plenty” trick (pulling huge amounts of food out of a small container) requires hiding things in the magician's robes; as in the “loaves and fishes” stories which Mark does have, the point of the “water into wine” story is that the quantity was too great for such an explanation.