Was Peter ever in Rome? Is there any Biblical or historical evidence that Peter ever went to Rome? If not, how can Peter be considered the first Pope?

Please note that unlike the following questions (What evidence is there that Peter was a bishop in Rome? and What exactly is Peter's connection to Rome according to Catholic dogma?), this one does not limit responses to biblical proofs and to 1st century evidence or asks how St. Peter is connected to Catholic dogma.
Biblical

The only Biblical information we have that Peter might have been in Rome is the cryptic reference in 1 Peter 5:13, "She who is in Babylon, chosen together with you, sends you her greetings, and so does my son Mark."  If "Babylon" is the cryptic name for Rome, as is commonly claimed, then this might be the sole source of information from the Bible.

Historical

The extra-biblical data about Peter being in Rome is highly debated and scant at best.  This author,  Otto Zwierlein, claims that there is no reliable evidence that Peter was ever in Rome - see the following document: Has St. Peter ever been in Rome?

Wikipedia has a short discussion about this but produces no solid evidence of Peter in Rome. It concludes:


  Clement of Rome' First Letter, a document that has been dated from the
  90s to the 120s, is one of the earliest sources adduced in support of
  Peter's stay in Rome, but Zwierlein questions the text's authenticity
  and whether it has any knowledge about Peter's life beyond what is
  contained in the New Testament Acts of the Apostles. The letter also
  does not mention any particular place, only saying: "Peter, through
  unrighteous envy, endured not one or two, but numerous labours and
  when he had at length suffered martyrdom, departed to the place of
  glory due to him." (ch. 5) A letter attributed to Ignatius of Antioch
  to the Romans might imply that Peter and Paul had special authority
  over the Roman church,[21] telling the Roman Christians: "I do not
  command you, as Peter and Paul did" (ch. 4), although Zwierlein says
  he could be simply referring to the Epistles of the Apostles, or their
  mission work in the city, not a special authority given or bestowed.
  Zwierlein has questioned the authenticity of this document and its
  traditional dating to c. 105–10, who says it may date from the final
  decades of the 2nd century instead of from the beginning.


In short, there are only hints that St. Peter might have been in Rome but almost no historical, unequivocal evidence.  At the same time, none of the historical data precludes St. Peter being in Rome near his end.