I would highly recommend John Julius Norwich’s excellent book “The Popes” as an enjoyable and informative read.

Unfortunately for your question, for the early Popes JJN states that “there are still too many question marks for any confident deductions to be made.” (p6).

What can be said? St Irenaeus of Lyons wrote a list of the first thirteen Popes from St Peter to his contemporary Eleutheris (c175-89).

However, until the 9th century the title “Pope” was applied generally to any senior member of the community and may not specifically have been to the Bishop of Rome. It is speculated that Peter did not found the church in Rome but was subsequently “made” the “first” Bishop of Rome as an honorific / way of later Popes trying to build prestige for themselves following the ascendancy of Rome as the head of the Catholic Church (something not in any way seen as likely in the first few hundred years of Christianity).

For the first few years Rome was a backwater for Christianity with the focus remaining in the more Greek-centric Levantine; we know more about the bishops of Smyrna and Antioch at this time as they were the leaders of the Church in its early years.

Furthermore, a treatise called the Shepherd of Hermas, written in Rome at the beginning of the second century, speaks of “Rulers [plural] of the church”. JJN goes on to say “it is hard to say who was the first true Pope, or supreme bishop [of Rome]; but the process seems to have been complete by the time of Anicetus (c155-66)” whom he notes as the 11th Pope.

So, facts about Linus: probably very little. Speculation about him: he was likely one of the actual founders of the church in Rome and would have been one of its head when Peter visited Rome (if he did) but he was not a bishop in the way that there were Bishops in the Middle East or as we now think of them.

For other early Bishops of Rome, we have Clement, who often appears as third or fourth in the list of Popes, and appears to have known Peter personally. Though all we have is a letter of his from Rome to Corinth from c96AD.

Unfortunately we need to look to the 2nd century onwards for real knowledge about Bishops of Rome.

For fun Pope Facts, you may wish to learn about the execution of Pope Formosus (after he had already died, buried and exhumed), Marozia (“lover, mother and grandmother of Popes”) or “The Great Pornocracy” under Pope John XII.

Apologies for any typos, this is written on my phone.

Precious little, and nothing certain.

Irenaeus, in Against Heresies 3.3.3 (ca. 180 CE), provides a succession list for Rome (the episcopacy, let's remind ourselves that the institution of the Papacy developed over time). The list goes:

Linus, Anacletus, Clement, Evaristus, Alxander, Sixtus, Telephorus, Huginus, Pius, Anicetus, Ster, Eleutherius.

There are a number of significant issues though. Firstly, these kinds of succession lists (for Rome, but also in other places trying to claim an apostolic lineage) do not appear until the late 2nd century CE. There is no earlier attestation of these lineages. Secondly, Irenaeus does not tie Linus to Peter, but to the apostles, Peter and Paul; the idea of that Peter was bishop at Rome is itself not abundantly clear. Thirdly, although Irenaeus refers this Linus to the one mentioned 2 Tim 4:21, there's nothing beyond the name to associate them.

It is true that Irenaeus could have accessed Roman church traditions on a succession lineage, but he would still be the earliest mention of it. It's not helped by considering our other, later, sources. Eusebius follows Irenaeus, but other traditions appear to follow Hippolytus, for example the Liber Pontificalis (a possible 6th century text), which lists Peter, Linus, Cletus, Clement, Anacletus. It also states that Peter ordained two bishops, Linus and Cletus (Liber Pontificalis, 1.3).

Apostolic Constitutions gives us Linus ordained by Paul, and Clement as 2nd, ordained by Peter.

And yet, further complications need to be acknowledge - Clement is mainly famous for the letter of 1 Clement, which (a) is attributed to him but contains no direct acknowledgement; (b) it's unclear that Clement was a bishop at Rome; If I recall Wellborn's argument on this correctly, he suggests that Clement was more likely the church secretary at Rome, tasked with writing outgoing correspondence.

Moreover, the idea of 'monarchical episcopacy', that a single bishop held power and authority in a city, does not appear to be the case so early, not in Rome and not elsewhere. Just as the notion of 'pope' is in great danger of anachronism when read back into this early period, even 'bishop of Rome' is disputable.

Alas, we know so little that it is difficult to be certain of anything. Was Linus a real person? Probably. Was he a sole-bishop at Rome? Less likely. Did this occur as a direct result of Peter handing him apostolic authority? Even less likely.