The birth of the Church was at Pentecost and Peter was definitely present and immediately made his presence felt as the vicar of Christ on Earth subsequent to the descent of the Holy Spirit on him and the other apostles and Our Lady in the upper room.
Then Peter stood up with the Eleven, raised his voice, and proclaimed to them,
Acts 2:14 NABRE
But, the Pope, is the Bishop of Rome and after Peter's marvelous martyrdom, St. Linus was chosen as the next leader.
The blessed apostles, then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate.
St. Irenaeus - Against Heresies 3.1.3
and after him came Cletus and Clement etc... until Francis and the collapse of humanity as we know it.
Oh, but I'm getting ahead of myself and so are you in this question.
I think the best sense to read some of what Jesus says to Peter is in the anagogical (eternal) sense. Jesus even talks about Peter's death before it happens. There's no reason to think He's not also going to talk about his ministry before it happens. And after Peter's death, a lot of sacred scripture would have very little meaning if it didn't also apply in a very real sense to today and until the (imminent) end of the world.
So, I think you're right on all the points you make about when Peter's ministry begins. When you want to call him Pope is up to you. If you're talking about when he was able to proclaim Ex Cathedra, keep in mind he didn't have a chair to proclaim from. And beyond that, even today, it's the Pope or the Pope in Communion with the Bishops, who make infallible magisterial teachings.
So, back then, Peter, Clement, John and James the less (as well as any other Apostles and Bishops in town) could have gotten together to figure out what to do with all the power they'd been given by Christ and all the trouble they were given by the emperors.