ND, Peter's celibacy isn't a Roman Catholic claim. Celibacy in the Roman priesthood, and the bishopric by extension, came much later (and is a point of dispute, actually, between our two Churches, but one of the more minor ones).
As for the "how" it came about...I've explained how the bishopric is actually the original office above. I've also pointed out that Rome had primacy from the very beginning. This primacy, though, in the early days couldn't be absolute, because, well, conditions forbade it. Rome was, after all, quite hostile to Christianity.
Now on the rest of this, though, I'm going to remain as neutral as I can. Please note that I have a strong bias, and it is a complex issue.
I'm going to skip a few years, well centuries. When the Emperor Constantine made Christianity equal in standing to paganism (not yet the official religion...that comes later), he also moved the capitol of the Empire to New Rome. When he did, he left Old Rome in the hands of the bishop of Rome. Hence, the Roman Bishop gained a degree (far from absolute) of power over the city of Rome, and this was unique in this way among all other bishops.
Further, in Constantine's reign, we have the Council of Nicea. This, like the Council of Jerusalem, had enough bishops to come very close to including the whole of the Christian world (a very decentralized world still), including bishops from outside the Empire (sorry to those who think Constantine corrupted the Church, but his reach didn't extend that far, and many of these same bishops were tortured for their faith. I doubt they ducked and cowed just because another Emperor came to the throne). The relavent portion of Nicea here is that it placed limitations on the bishops that they may not interfere in other jurisdictions (I Clement is just such a case, where the Roman Church wrote the Corinthian Church over their problems). I'm not sure if this is the council that placed the Roman Bishop in a place of special arbitration, but I know it is in effect later and will come sharply into play.
Well, next I skip to the fifth century and the Council of Ephesus. A man named Nestorius began teaching that Mary wasn't the Theotokos (this is translated literally both as "Bearer of God" and "Mother of God"). Naturally, this caused quite a stir, and there was even fighting in the streets. Nestorius raised a band of ruffians, and one subsequently alligned itself with St. Cyril. Cyril taught that Jesus had one nature, which was both divine and human. He qualified this, however, with phrases like "without change" or "without alteration," to imply that neither nature lost anything. Cyril, consequently, won out in the Council of Ephesus. As an aside, this is one of the most tragic moments in Church history, for it is in this period we see the most bitter bickering, the slaughter of Hypatia, and the first great schism of Christian history at the Council of Chalcedon, and they comprise the "Oriental Orthodox" today.
Well, like all things, people over-corrected on the mystery of Christ. The next teacher taught what St. Cyril taught...except he omitted the qualifying phrases of Cyril, and thus made Jesus a hybrid. Again, an Ecumenical Council had to be called. Here, they settle the issue, but there is an odd way in which they go about it. Pope Leo sent his dotrinal statement, called "The Tome of Leo" as a definition of his beliefs. The Council poured over it and found it acceptable, they proclaimed "Peter has spoken!" and settled the issue. The Orientals then split off of the rest of the Church and comprise several groups, such as the Indian, the Ethiopian, the Coptic, the Assyrian, the Syriac, etc. Orthodox Churches. At most, though, there are two churches here. Don't let the ethnicities fool you. It's a means of orginization without centralization. The Coptic Church still maintains its "Pope," while the Orthodox (not Oriental) bishop of Antioch also presides. It gets confusing
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Not long after this, the Western Empire fell. Rome continued in the East, and Constantinople continued as her capitol (there never was a "Byzantine Empire." That is a racist slur created by French and German historians IMO). As a result, the Roman people in the West were under rule by Germanic Arians (not to be confused with Aryans). The Pope, then, acted as a sort of safeguard, and a means by which the Church, and their culture, might continue. The same thing would happen in the East when the Roman Empire was finally destroyed by the Turks. This placed the Pope in a place of preserving his culture, protecting his people, and protecting doctrine. The Germanic peoples were a real threat, and they could (and would) do many things, so they had reason to fear.
The next major event is the Sixth Ecumenical Council. A new heresy arose...this time saying that Jesus' human will was swallowed up by His divine will, just as a drop is swallowed up in the ocean. Here, though, Pope Honorius was actually condemned a heretic. He recanted, though, very quickly, and submitted to the Council. This is also the last Council I will digress over, of the Seven (and some in Orthodoxy say Nine) Ecumenical Councils. I believe that the Roman Catholic Church has twenty-seven. I may be in error here.
One interesting thing to note, from my vantage-point, is that Rome didn't call the Councils. They were all called by Emperors. Rome, like most other bishops, inevitablly approved them all in the end. Rome's approval was important, though, because with the exception of Honorius, it had remained steadfastly Orthodox. It is also at this point I must tread very carefully. I enter a controversial era between my Church, the Orthodox Catholic Church, and the Roman Catholic Church. I will try to compose my words here with care, and I won't say "who was right" in 1054, though everyone knows my opinion
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I also need to continue this in the next post. It may well be too long already.
