I don't know when St. Peter was called bishop of Rome first. In apostolic time that terminology was quite unsettled. It is clear, from the references I gave you that he ended up in Rome and had unique authority given by Christ. As for these historical trivia you should find someone else to ask. I am here to defend Catholic Christianity, not play Jeopardy.
No, St. Peter was not made pope in Matthew 16, as Christ spoke in future tense then. The suggestion made by Peter to obtain power echoes Satan's but was done innocently; he followed up on the night of the betrayal by drawing a sword. His worst act is, of course, his denial of Christ. None of these impedes his status following the charge to feed and guide the sheep given after the Resurrection.
I find Peters rejection, repentance and redemption particularly encouraging. What a loving, merciful God we have!