Is there any extra-biblical written evidence that any of the Apostolic Fathers (i.e., the early Christian theologians of the 1st and 2nd centuries) knew the Apostle Paul? If so, what is the written evidence that they were taught by Paul or even heard him preach? Are any of them mentioned in the Bible?
(Apparently, there is some discussion as to whether Linus and Anacletus were essentially “auxiliary bishops” and Clement was Peter’s actual successor.)
In any event, scholars agree that this Clement is the author of the (First) Epistle of Clement to the Corinthians. (A second letter to the Corinthians was also once attributed to him, but scholars now agree that the attribution is spurious.)
A certain Clement is named in Philippians 4:3:
Yes, I ask you also, true companion, help these women, who have labored side by side with me in the gospel together with Clement and the rest of my fellow workers, whose names are in the book of life.
There is not 100% certainty that this is the very same Clement who was the Bishop of Rome and wrote the epistle, but that is the traditional understanding. Moreover, it is plausible, given that the style of Clement’s epistle to the Corinthians is markedly similar to Paul’s (and the Church in Corinth was founded by Paul, which would explain why Clement was interested in what went on there). Also, Paul likely wrote the letter to the Philippians in Rome (see, e.g., Phil. 4:22), and so the connection between this Clement and Rome is plausible.
If that is the case, then Clement of Rome, an Apostolic Father, would have known St. Paul.
We don't know. One writing included among those attributed to the Apostolic Fathers is the Epistle of Barnabas. If the Epistle of Barnabas was written by Barnabas, then yes - Barnabas knew Paul.
An ancient tradition holds that St. Peter, while en route to Antioch to meet St. Paul, appointed Ignatius to be bishop. Since Paul spent time in Antioch and Ignatius became its bishop, they may have met. See http://maryourmother.net/Ignatius.html