Responsibilities of the Bishop of Rome:
http://chiesa.espresso.repubblica.it/articolo/1341814bdc4.html?eng=y
Every Pope as Peter’s successor, by authority of Jesus, but it is not necessary that the Pope always needed to reside in Rome from the beginning of the Church. However Rome became a natural center as it was head of the Roman Empire and that centrality remain in history to the present day.
As Bishop, he has that responsibility but he is above all, responsible to follow Peter as leader of the Church as His primary responsibility
http://www.vatican.va/archive/ccc_css/archive/catechism/p123a9p4.htm
874 Christ is himself the source of ministry in the Church. He instituted the Church. He gave her authority and mission, orientation and goal:
880 When Christ instituted the Twelve, "he constituted [them] in the form of a college or permanent assembly, at the head of which he placed Peter, chosen from among them."398 Just as "by the Lord’s institution, St. Peter and the rest of the apostles constitute a single apostolic college, so in like fashion the Roman Pontiff, Peter’s successor, and the bishops, the successors of the apostles, are related with and united to one another."399
881 The Lord made Simon alone, whom he named Peter, the “rock” of his Church. He gave him the keys of his Church and instituted him shepherd of the whole flock.400 "The office of binding and loosing which was given to Peter was also assigned to the college of apostles united to its head."401 This pastoral office of Peter and the other apostles belongs to the Church’s very foundation and is continued by the bishops under the primacy of the Pope.
882 The Pope, Bishop of Rome and Peter’s successor, "is the perpetual and visible source and foundation of the unity both of the bishops and of the whole company of the faithful."402 "For the Roman Pontiff, by reason of his office as Vicar of Christ, and as pastor of the entire Church has full, supreme, and universal power over the whole Church, a power which he can always exercise unhindered."403
A list of the first 10 popes as available at this website:
https://stpeterslist.com/the-first-10-popes-of-the-catholic-church
List of popes in the Church’s history
http://www.newadvent.org/cathen/12272b.htm