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Of course it is claimed that Peter was the pope from the time the "keys" were given to him.

Peter was surely chosen by Christ as the first pope at the interchange in Matthew 16:18-19, but the promise of the keys is in the future at that point. Certainly there was no conclave to designate St. Peter pope. Also, the congruence between the see of Rome and papacy has not been determined at that time, so while one may say that St. Peter was pope certainly by the time resurrected Christ charged him with His sheep in the spiritual sense, he did not become bishop of Rome till much later.

It is clear from the Acts that St. Peter was a figure exercising leadership and authority even if the Jerusalem council was presided by the local bishop, St. James. St. Peter also refers to his office as something coming from Christ and that he intends to perpetuate in 2 Peter 1-2.

As to the rest of your doubts regarding Peter as pope, I refer you to a few very clearly written articles.

There is ample evidence in the New Testament that Peter was first in authority among the apostles. Whenever they were named, Peter headed the list (Matt. 10:1-4, Mark 3:16-19, Luke 6:14-16, Acts 1:13); sometimes the apostles were referred to as "Peter and those who were with him" (Luke 9:32). Peter was the one who generally spoke for the apostles (Matt. 18:21, Mark 8:29, Luke 12:41, John 6:68-69), and he figured in many of the most dramatic scenes (Matt. 14:28-32, Matt. 17:24-27, Mark 10:23-28). On Pentecost it was Peter who first preached to the crowds (Acts 2:14-40), and he worked the first healing in the Church age (Acts 3:6-7). It is Peter’s faith that will strengthen his brethren (Luke 22:32) and Peter is given Christ’s flock to shepherd (John 21:17). An angel was sent to announce the resurrection to Peter (Mark 16:7), and the risen Christ first appeared to Peter (Luke 24:34). He headed the meeting that elected Matthias to replace Judas (Acts 1:13-26), and he received the first converts (Acts 2:41). He inflicted the first punishment (Acts 5:1-11), and excommunicated the first heretic (Acts 8:18-23). He led the first council in Jerusalem (Acts 15), and announced the first dogmatic decision (Acts 15:7-11). It was to Peter that the revelation came that Gentiles were to be baptized and accepted as Christians (Acts 10:46-48).
Peter and the Papacy
Tertullian, in The Demurrer Against the Heretics (A.D. 200), noted of Rome, “How happy is that church . . . where Peter endured a passion like that of the Lord, where Paul was crowned in a death like John’s [referring to John the Baptist, both he and Paul being beheaded].” Fundamentalists admit Paul died in Rome, so the implication from Tertullian is that Peter also must have been there. It was commonly accepted, from the very first, that both Peter and Paul were martyred at Rome, probably in the Neronian persecution in the 60s.
In the same book, Tertullian wrote that “this is the way in which the apostolic churches transmit their lists: like the church of the Smyrnaeans, which records that Polycarp was placed there by John; like the church of the Romans, where Clement was ordained by Peter.” This Clement, known as Clement of Rome, later would be the fourth pope. (Note that Tertullian didn’t say Peter consecrated Clement as pope, which would have been impossible since a pope doesn’t consecrate his own successor; he merely ordained Clement as priest.) Clement wrote his Letter to the Corinthians perhaps before the year 70, just a few years after Peter and Paul were killed; in it he made reference to Peter ending his life where Paul ended his.
In his Letter to the Romans (A.D. 110), Ignatius of Antioch remarked that he could not command the Roman Christians the way Peter and Paul once did, such a comment making sense only if Peter had been a leader, if not the leader, of the church in Rome.
Irenaeus, in Against Heresies (A.D. 190), said that Matthew wrote his Gospel “while Peter and Paul were evangelizing in Rome and laying the foundation of the Church.” A few lines later he notes that Linus was named as Peter’s successor, that is, the second pope, and that next in line were Anacletus (also known as Cletus), and then Clement of Rome.
Clement of Alexandria wrote at the turn of the third century. A fragment of his work Sketches is preserved in Eusebius of Caesarea’s Ecclesiastical History, the first history of the Church. Clement wrote, “When Peter preached the word publicly at Rome, and declared the gospel by the Spirit, many who were present requested that Mark, who had been for a long time his follower and who remembered his sayings, should write down what had been proclaimed.”
Lactantius, in a treatise called The Death of the Persecutors, written around 318, noted that “When Nero was already reigning (Nero reigned from 54–68), Peter came to Rome, where, in virtue of the performance of certain miracles which he worked by that power of God which had been given to him, he converted many to righteousness and established a firm and steadfast temple to God.”
These citations could be multiplied. (Refer to Jurgens’ books or to the Catholic Answers tract Peter’s Roman Residency.) No ancient writer claimed Peter ended his life anywhere other than in Rome. On the question of Peter’s whereabouts they are in agreement, and their cumulative testimony carries enormous weight.
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evidence had mounted to the point that Pope Paul VI was able to announce officially something that had been discussed in archaeological literature and religious publications for years: that the actual tomb of the first pope had been identified conclusively, that his remains were apparently present, and that in the vicinity of his tomb were inscriptions identifying the place as Peter’s burial site, meaning early Christians knew that the prince of the apostles was there. The story of how all this was determined, with scientific accuracy, is too long to recount here. It is discussed in detail in John Evangelist Walsh’s book, The Bones of St. Peter. It is enough to say that the historical and scientific evidence is such that no one willing to look at the facts objectively can doubt that Peter was in Rome. To deny that fact is to let prejudice override reason.
Was Peter in Rome?

I was, however wrong to say that St. Mark was martyred in Rome. I was thinking of St. Paul. Mark was martyred in Alexandria. If you are interested in the relationship between St. Peter and St. Mark, (whom St,. Pater called his "son", although possibly he meant godson), see St. Mark