All right Izdaari,
Whether we go off legend (Peter) or accepted historical fact (Leo I, based on the universal jurisdiction argument) I think we can agree, that in either case, it still was not anybody named "Jesus."
B.
Well, of course! Jesus used several different titles for himself, Son of Man being probably the most common, but Pope was definitely not one of them.
However, the Christian church existed and began to articulate its beliefs not long after the resurrection of Jesus. The surviving apostles along with Jesus' brother James, and Paul who also became accepted as an apostle, were collectively the authorities of the church. There was at that point no Pope. Of course, as one of the original Twelve, Peter was highly respected, but was not top dog by any means. At least, he accepted a public rebuke on a doctrinal matter from Paul, and didn't try to assert any authority. Nor was he accepting Paul as a superior authority. He accepted correction from Paul as an equal, being convinced that Paul had the right of the matter.
OTOH, if you're talking about the Roman Catholic Church, and not the catholic (i.e. universal) church, I agree that didn't exist until much later. I would say it didn't exist until Leo I. But of course I'm not a Catholic. I would expect Catholics to insist that Peter was the first Pope. The problem with that is that until Leo I, the Bishop of Rome didn't have authority over the other Bishops. Instead, like Peter himself, the office was respected as "first among equals."