Roman governors did not convert to Christianity until after they resigned from their office, because it was part of their official duty to sign death warrants. Government was considered incompatible with Christian belief during the pre-Constantine period.
The Catholic Church is supposed to be based on historic precendent, but that's what the Pope taught in the 2nd century AD.
1) That isn't what it says today.
2) The infallibility of the pope only applies to ex-cathedra statements. Besides, I'd find it rather surprising to have a Roman Governor publically convert to Christianity in the 2nd Century AD.
3) Christ ordered his apostles to carry swords. I doubt they were for decoration. The early Popes were certainly aware of this fact.