No, the UNSHAKABLE rock is Jesus Christ - always was always will be. When St. Peter died a martyr's death in 64 A.D. that "first place" of leadership did as well. You have NOTHING in Scripture that proves Peter's leadership among the Apostles and the local assemblies he helped to form was, or could be, passed on from Peter to someone else. Your religion takes a huge logic leap - unfounded within the writings of even the early church leaders - by saying this "primacy" was something that could be handed down for thousands of years and was what Jesus meant by saying to Peter what He did.
As has been shown at least a dozen times on this thread alone, there was no Pope Peter, head over all Christendom, there was no Pope of Rome nor a succession of Popes of Rome that led all of Christendom, there was no head bishop that all Christendom had to obey. There was no church called the Roman Catholic Church until at least four centuries after the last Apostle died. The very word "catholic" wasn't even used until the second century and then only as an adjective describing the "universal" body of believers in Christ. All that Roman Catholicism teaches today about Apostolic Succession is something they invented and developed over time. They didn't get it from Christ, they didn't get it from Peter.
sure they did. If you read he bible and interpret it PROPERLY it is very easy to see.