Dear sister in Christ:
I'll try to respond to your questions as well as I can--and as briefly as I can!
First, on Peter and Rome. It is quite clear that Peter did not found the church in Rome. When Paul wrote to the Romans, he did not even mention Peter--which he did when he wrote to other churches where people know who Peter was. When he arrived in Rome, there were already Christians there who came to meet him. So the origins of the church in the city of Rome are lost in the night of history, and must probably be attributed to some of the many anonymous Christians who did and still do so much for the expansion of Christianity.
That Peter was in Rome, and probably died there, is sufficiently attested by ancient writers as to be at least believable, and even likely. That he was a "bishop" is very unlikely, for there is very little evidence--if any---that the apostles ever called themselves bishops. Furthermore, if he arrived in Rome when there already was a church there, it is to be imagined that the church in Rome already had its leadership.
Now to the other question: When was the church in Rome the "Roman Catholic church"? In a way, probably since its beginning. The term "catholic" simply meant orthodox and holding to the faith of the church everywhere. Thus, the church in Ephesus was "the catholic church of Ephesus," and the church in Antioch was the "catholic church of Antioch," and in the same way the church in Rome was catholic.
On the other hand, if your question is when the church in Rome came to have jurisdiction or power over the rest of the church, then the answer is that this was a long process. The process was simply the process that we see even today, where churches in cities and centers of power tend to have more authority that the smaller churches in smaller towns. All over the roman Empire, the churches in provincial capitals eventually gained jurisdiction over an area. (The term "diocese" was a jurisdiction within the Roman Empire, and is now the jurisdiction of a bishop.) This was true of Rome, as well as of Alexandria, Antioch, Carthage, etc.
What happened in the Latin-speaking West was that the Empire disappeared as a result of the invasions of the Germanic "barbarians." The power vacuum was such that someone had to fill the gap and try to bring some order. Thus the church of Rome, which took the leadership in that process, became the center of authority for the entire Western church. This was a process that began late in the fourth century, and took great leaps with Leo the Great in the fifth and Gregory the Great at the end of the sixth.
I hope this answers your questions. Blessings on you.
Justo Gonzalez