That's a good point, but when you look at the geography of the area to which he writing [1 Peter 1:1-2].....and to whom he's writing, the answers are obvious. Babylon is no more cryptic that Pontus, Galatia, Cappadocia, Asia and Bithynia.
The Israelites had been placed on the north and west borders of Assyria primarily to act as a buffer. It would not be strange at all to find some of them 700 years later somewhat north and west of there, now settled on the shores of the Black Sea.
No, Peter is speaking of Babylon on the Euphrates. There is no evidence that Rome was ever referred to as Babylon until John wrote of it in Revelation. We are in agreement. This would have been long after Peter had died and there is nothing in his letter that appears cryptic.....or apocalyptic. Peter is blunt to the point of being extremely plain. There is nothing mystical about this letter and he is very direct in stating his message.
In A.D. 40, three years after Paul's conversion, Peter is still hanging around Jerusalem [Galatians 1:18]. He is imprisoned (by Herod Agrippa 1) but miraculously escapes [Acts 12:5-11] and Herod dies [Acts 12:23] in the fourth year of the reign of Claudius (A.D. 45). Paul again visits Peter in Jerusalem [Galatians 2]....14 years later and they agree on who should evangelize who. Paul to the Gentiles and Peter to the Israelites. It is now A.D. 54 and as far as I can tell Peter hasn't traveled too far! Peter does go to Antioch soon after where Paul had established his headquarters [Acts 15:35] and they have their famous meeting [Galatians 2:11-14].
From then to about A.D. 60 Peter evidently made a evangelistic journey throughout the eastern areas....taking along his wife [1 Corinthians 9:5] and it is assumed that Peter stayed in the east (Babylon) since about the same time Paul is writing from Rome, saluting 27 different people and Peter is absent....and is not even mentioned [Romans 16]. Paul probably doesn't allude to him at all because he has never been there. The church was Gentile and he would have had no business there to begin with.
Paul finally gets to Rome about 63 A.D., writes four epistles (Ephesians, Philippians, Colossians and Philemon) and never mentions Peter. He states in [2 Timothy 4:11] that only Luke is with him. This is during his second trip to Rome about 67/68 A.D. shortly before his death.
We are not in agreement at all. First of all, I don't define 30-odd years as "long after Peter died". Peter died in around 66-7, John was writing mid-90s. And Peter's letter was addressed specifically to the churches of Asia Minor...where John was. So I don't have any doubt that John was very familiar with Peter's letter--and it would be as natural as anything for John to pick up on Peter's reference to Rome as Babylon and expand on it in Revelations. The "fact" that Rome was not mentioned as Babylon earlier than Revelations is predicated on the resolution of this question in 1 Peter: if it is Rome, then the first-mention-in-Revelations position obviously can't be sustained.
As to the non-apocalyptic tone of Peter's letter, he didn't have to be writing an apocalypse in order to do a simple play on the city. He may have been cryptic about the city name for a reason...perhaps persecution.
Paul probably doesn't allude to him at all because he has never been there. The church was Gentile and he would have had no business there to begin with.
Paul says in Romans 15:20 that he has been "hindered from coming" to them because his aim was "to preach the gospel, not where Christ was already named, that I might not build upon another man's foundation." If a) the Church in Rome was solely Gentile and b) Paul was the only Apostle to the Gentiles, then how could it be that an Apostle was there founding a church before him? Seems to me somebody besides Paul had business there, and probably somebody big by the sound of it.
And as to why Paul does not salute Peter in Romans, Peter may well have been galavanting around and not in the city. Priscilla and Aquila left when Claudius expelled the Jews...Peter may have as well. It is pretty clear (to me anyway) that Peter was not in Rome the whole time. He made trips to other places, as you mention. That in no way contradicts the fact that he came to Rome and that he died there.