Can apostolic succession be historically proven? Yes.
Where did bishops originate?
Ultimately (in conception) from the apostolic deposit and Holy Scripture. I wrote in my first book, A Biblical Defense of Catholicism (1996):
In the Septuagint (the Greek translation of the Old Testament), episkopos is used for overseer in various senses, for example: officers (Judges 9:28, Isaiah 60:17), supervisors of funds (2 Chronicles 34:12,17), overseers of priests and Levites (Nehemiah 11:9, 2 Kings 11:18), and of temple and tabernacle functions (Numbers 4:16). God is called episkopos at Job 20:29, referring to His role as Judge, and Christ is an episkopos in 1 Peter 2:25 (RSV: “Shepherd and Guardian of your souls”).
Are bishops successors of the apostles?
Yes. The classic biblical argument is the replacement of Judas with Matthias. Judas was actually called a bishop in Acts 1:20, which is part of the passage describing the succession (“office” in RSV but “bishopric” in KJV and the usual word for “bishop”: episkopos). Moreover, Eusebius wrote:
All that time most of the apostles and disciples, including James himself, the first Bishop of Jerusalem, known as the Lord’s brother, were still alive . . . (History of the Church, 7:19, translated by G. A. Williamson, Baltimore: Penguin Books, 1965, 118)
James is called an apostle by St. Paul in Galatians 1:19 and 1 Corinthians 15:7. That James was the sole, “monarchical” bishop of Jerusalem is fairly apparent from Scripture also (Acts 12:17; 15:13, 19; 21:18; Gal 1:19; 2:12).
Did these early bishops (and/or apostles hand-pick their successors?
They certainly did in the instance of Matthias, though it was a “collegial” decision (determined specifically by casting lots). In like fashion, St. Paul appears to be passing on his office to Timothy (2 Tim 4:1-6), shortly before his death, around 65 A.D.
In what sense can we say that St. Peter was pope?
The early Christians even during apostolic times right after Pentecost recognized Peter as the leader and focal point in the Church. The late great Protestant scholar James D. G. Dunn stated, along these lines:
It is Peter who becomes the focal point of unity for the whole Church – Peter who was probably the most prominent among Jesus’ disciples, . . . Peter who was the leading figure in the earliest days of the new sect in Jerusalem, . . . he became the most hopeful symbol of unity for that growing Christianity which more and more came to think of itself as the Church Catholic. (Unity and Diversity in the New Testament, London: SCM Press, 2nd edition, 1990, 385-386)
Was St. Peter the first “monarchical” bishop of Rome?
Dionysius, writing to the bishop of Rome around A.D. 70, is an early historical witness:
8. And that they [Peter and Paul] both suffered martyrdom at the same time is stated by Dionysius, bishop of Corinth, in his epistle to the Romans, in the following words: You have thus by such an admonition bound together the planting of Peter and of Paul at Rome and Corinth. For both of them planted and likewise taught us in our Corinth. And they taught together in like manner in Italy, and suffered martyrdom at the same time. I have quoted these things in order that the truth of the history might be still more confirmed. (Eusebius, History of the Church, Book II, 25: 5,8)
F. F. Bruce, the renowned Protestant New Testament scholar observed:
As for Peter’s association with the Roman church, this was . . . reflected in the greetings sent to the readers of 1 Peter from the church (literally, from “her”) “that is in Babylon, elect together with you” (1 Pet. 5:13) – if, as is most probable, Babylon is a code-word for Rome. (Peter, Stephen, James, and John, Grand Rapids, Michigan: William B. Eerdmans Pub. Co., 1979, p. 44)
The claim that Peter and Paul were joint-founders of the Roman church – attested, as we have seen, by Dionysius of Corinth – is earlier than the tracing of the succession of bishops of Rome back to them, which is first attested in Irenaeus but may go back to Hegesippus [Irenaeus, Against Heresies 3.3.1-3; for Hegesippus see Eusebius, Hist. Eccl. 4.22.3]. (Ibid., pp. 46-47)
Did St. Peter choose his successor?
St. Irenaeus says that he did do so (with St. Paul):
The blessed apostles, having founded and built up the church [of Rome] . . . handed over the office of the episcopate to Linus. (Against Heresies 3:3:3 [A.D. 189])
Thus we have a very eminent Church father (Irenaeus) writing around 189, confirming that Peter picked his successor as pope. That was only about 120 years earlier. It would be like us writing today about Theodore Roosevelt. It's not that long.
I'm 61 years old. Up until 1969 and 1983, I could talk to my two grandfathers, who were born in 1891 and 1893. Thus I was only one eyewitness away from people who were alive in 1900: 120 years ago. That was like (older) St. Irenaeus in relation to St. Peter at the end of his life. The Church had writing and it also had far stronger and more reliable oral traditions than we have today.
Who were St. Peter’s first two successors as pope?
Eusebius, the first Church historian, confirms that Linus was the first successor, and Clement the second (third pope):
Paul testifies that Crescens was sent to Gaul [2 Tim. 4:10], but Linus, whom he mentions in the Second Epistle to Timothy [2 Tim. 4:21] as his companion at Rome, was Peter’s successor in the episcopate of the church there, as has already been shown. Clement also, who was appointed third bishop of the church at Rome, was, as Paul testifies, his co-laborer and fellow-soldier [Phil. 4:3]. (Church History 3:4:9–10 [A.D. 312]).
Can apostolic succession be historically proven?
Sure. It is precisely a claim to actual history: that can be analyzed and examined via the usual historiographical methods. The apostolic deposit and apostolic succession were passed down in history, in an unbroken chain.