No one seems to be responding to your question, so I’ll give it a shot. First things first. If you want a very good history of Christianity written for general readers you can do no better than Diarmid MacCulloch’s excellent Christianity: The First Three Thousand Years (2009) from which comes some of what I write here. If you want to know more about the early popes, Eamon Duffy’s Saint and Sinners: A History of the Popes (1997) is a very readable introduction.

So, when and where was the Roman Catholic Church founded? That’s a slippery question. Historians of early Christianity now typically speak of “Christianities” in the early days of the Christian church. There were dozens of small Christian communities scattered across the Mediterranean world in the first centuries after the death of Jesus (conventionally in c. 30-33 CE, again, not BC) and they did not agree on points of doctrine, worship rituals, or what texts were to be considered the revealed word of God. In fact, some of them were branded as “heresies.” It’s congregations of the bishops, or “overseers,” of these various regional groups that over time hammer out a common doctrine and practice. For example, it’s only in the 200s that most of the books of what becomes the New Testament are widely agreed upon and only in 393 that a synod of bishops at Hippo in North Africa approves the “canonical” books of the NT as we know it today. There are some very interesting texts, some of them a little wacko, that got left out. There’s not really a single historical moment where we can say, “Voila, here is Christianity.” Your reference to the Council of Nicaea (in 325 CE, not BC) is certainly an important date insofar as the Nicene Creed approved there became one of the first well considered and lasting statements of the basics of Christian belief (as it still is today). In fact, it was convened largely to combat what we call Arianism, the most dominant heresy threatening “orthodox” belief in this period, which offered a different interpretation of how Jesus could be both God and man.

Despite tradition, Peter was decidedly not the founder of Christianity, nor, historically speaking, was he the first pope. From the gospel of Matthew and the Acts of the Apostles it seems that he held a revered place among early Christians; he is thought to have founded the church at Antioch. But Paul occupied a much more important place in honing the essential doctrines that became Christianity. Both Peter and Paul ended up in Rome—they seem to have been a “team of rivals”—where they were both martyred in the 60s CE. But there is no indication that there was a formal structure of bishops in Rome at this point, as there were in some Christian towns in the eastern Mediterranean. There were eventually bishops of Rome long after their deaths, in fact more than one at a time early on, but the first use of papa or pope starts to be used only at the beginning of 4th century. Frankly, we don’t know who the first pope was.

Why was the Vatican region chosen for the St Peter's Basilica? Because it was built over where Christians as early as the 2nd century believed his burial place to be. In around 160 a shrine was built at what was supposed to be Peter’s burial place (or perhaps where he was martyred). That spot is now under the confessio built at the foot of the main altar of St. Peter’s basilica underneath Bernini’s great baldacchino. In 1939, archaeological excavation actually found the shrine and bone fragments that may or may not be Peter’s. (Pope Paul VI declared them to be such, but that’s controversial.) In the 4th century the Emperor Constantine, the first emperor to convert to the new religion and, incidentally, the man who called the Council at Nicaea and presided over it, had a basilica built over that spot. The new 16th century-basilica replaced that older church, and incidentally helped spark the Reformation; many people thought the money collected to finance the rebuilding was excessive.

What's the relation between the Pentecost (St Peter's first sermon) and these locations, if there is any? There is none. The Acts of the Apostles (ch. 1) describes Pentecost but it happened in Jerusalem where the 12 apostles were said to be gathered with Jesus’s mother, Mary, and “about 120” others. The word itself is a Greek one, meaning the 50th day after Passover (a reminder that early “Christians” were still thinking very much in Jewish religious terms). Supposedly the Holy Spirit descended and inspired the apostles to begin speaking in different regional languages as both a guarantee of miraculous gifts the Holy Spirit bestows on the followers of Jesus and as a prompt to go forth and preach his message.

Finally, what's the difference between the Roman Catholic Church and the Apostolic Catholic Church? Effectively there is no difference. They're both names for the same thing. “Catholic” (it means “universal” in Greek) and “Apostolic,” along with “one” and “holy,” are descriptors used to define the church in the Nicene Creed. But you may be thinking of the Apostolic Catholic Church in the Philippines, which is sort of an offshoot of the traditional Catholic church although it does not recognize the Roman pope.