The issue here is Apostolicity and the succession thereof; from the Nicene creed (emphasis mine):
We believe in one holy catholic and apostolic church
The doctrine rests on the central tenants that in Matthew 16:18-20 Jesus invested apostolic leadership and kingdom authority in Peter and the apostles and Matthew 18:18 via the "keys of the kingdom of heaven" and the power to "bind" and "loose", and finally that Peter ended his ministry in Rome, bringing the principle seat of the church's authority to Rome when he died.
Matthew 16:18
17 Jesus replied, “Blessed are you, Simon son of Jonah, for this was not revealed to you by flesh and blood, but by my Father in heaven. 18 And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overcome it. 19 I will give you the keys of the kingdom of heaven; whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven.”
Matthew 18:18
18 “Truly I tell you, whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven.
That Peter came finally to Rome appears to be a settled question, according to the Catholic Encyclopedia:
As soon as the problem of St. Peter's coming to Rome passed from theologians writing pro domo sua into the hands of unprejudiced historians, i.e. within the last half century, it received a solution which no scholar now dares to contradict; the researches of German professors like A, Harnack and Weiasticker, of the Anglican Bishop Lightfoot, and those of archaeologists like De Rossi and Lanciani, of Duchesne and Barnes, have all come to the same conclusion: St. Peter did reside and die in Rome.
And Peter's martyrdom in Rome seems to be upheld by the early church fathers and it forms their own claim for apostolic authority; from the above linked article:
Beginning with the middle of the second century, there exists a universal consensus as to Peter's martyrdom in Rome; Dionysius of Corinth speaks for Greece, Irenaeus for Gaul, Clement and Origen for Alexandria, Tertullian for Africa. In the third century the popes claim authority from the fact that they are St. Peter's successors, and no one objects to this claim, no one raises a counterclaim.
It's seems, at least in my reading of early church writings, that the Roman church may have reached a place of significance within the wider church body by the 3rd or 4th century. The church in Rome is either called on or preemptively steps in to arbitrate disputes in other churches. The council of Sardica, 343, agreed that to "honor the memory of the Apostle Peter" they should "provide that letters be sent to Julius, Bishop of Rome" regarding any legal case involving a Bishop which is in dispute, for the Bishop of Rome to possibly hear the appeal. In 376 St. Jerome calls the "chair of Peter" the "Rock on which the Church is built". And in about 401 Pp. Innocent I claims the authority of the Roman See such that "any just pronouncement might be confirmed by the authority of [the Roman] See". (Documents of the Christian Church, in the section entitled The Authority of the Holy See).
Who is St. Linus and why did He succeed St. Peter when St. John was still alive?
The bishop Linus who succeeded Peter is believed to be the one mentioned in the letter to Timothy:
2 Timothy 4:21
20 Erastus stayed in Corinth, and I left Trophimus sick in Miletus. 21 Do your best to get here before winter. Eubulus greets you, and so do Pudens, Linus, Claudia and all the brothers and sisters.
Linus would have succeeded Peter as the bishop of the Roman congregation; John's responsibility was another church body.
Note that much of what we understand about the happenings of this time were lost in antiquity with only church traditions passing on details from one generation to the next. Several of the Roman persecutions made concerted efforts to wipe out Christianity, including burning as much of her writings as they could.
What is the basis for voting for a Pope?
The process by which the Pope is selected by the conclave of bishops has its roots in a simple consensus of church leadership which is believed to reflect the will of the Holy Spirit. It is based on the idea of wise counsel, the idea being that with many seeking the will of God on a particular matter, some might miss it but the majority will not. It is not supposed to be a political or popularity contest, but a genuine seeking of the Holy Spirit's leading. What actually happens in practice is hard to say (but I would certainly not base my perceptions on Hollywood depictions like Angels and Demons).
Why don't Popes do the same things as Peter (heal people, write epistles, etc)?
On the miracles, the question of why they are so relatively rare in the modern church is a separate issue, for a separate question.
As for epistles, this is precisely the big bone of contention between Catholic and Protestant Christians. The Catholic church teaches that the apostolic authority to teach with authority via the Holy Spirit continues to this day, via the Papal letters and encyclicals and exercise of the church's "magesterium" to pronounce doctrines as being infallible, making them dogmas (the latter being the only time, contrary to popular misconception, which the Pope is considered infallible).
Why is there a Pope if it isn't mentioned in the Bible (while apostles, pastors, etc. are)
Pope is simply the Latin word for "father", similar to and root of "papa" in other modern Romantic languages. Judeo-Christian culture has a long established history of referring to respected elders and teachers as "father", as a term of simple endearment and respect (actually, Asian cultures do this also). The moniker "Pope" simply stuck, over time, as the title for the Bishop of Rome. In fact, it's not clear to me if there is any difference in Italian for the general title for a priest, bishop or the Pope, other than perhaps capitalizing the 'P'.
