That hasn't happened yet, St. Peter's Basilica is a large Church but there are no facilities for someone to actually live there.
The church of the Pope as bishop of Rome is the Basilica of Saint John Lateran in Rome. The church of the Pope as Pope is Saint Peter's Basilica in the Vatican.
No, it is not a requirement.
In 2014 the current pope is Pope Francis.
The pope does not live in a chapel, but lives in the Apostolic Palace. The Sistine Chapel lies between the Apostolic Palace and St. Peter's Basilica.
The normal residence of the pope is in the Papal Apartments of the Apostolic Palace. However, Pope Francis has chosen to live in a much smaller apartment in the Casa Santa Marta which is also near to Saint Peter's Basilica.
Benedict is now retired and has nothing to do with the Basilica.
The Basilica of Saint Peter is a monument to the first pope - Saint Peter - who is buried under the basilica. It is used for special occasions - the canonization of saints, the funerals of popes, Christmas Midnight Mass, etc. It is not the church of the Pope. The Church of the Pope as the bishop of Rome is the Basilica of Sant John Lateran in Rome.
As pope, his church is St. Peter's Basilica in the Vatican but as Bishop of Rome, his church is the Basilica of St. John Lateran in Rome, Italy.
Yes, he is the leader of the Catholic Church. As bishop of Rome his church is the Basilica of Saint John Lateran. As Pope, his church is Saint Peter's Basilica.
San Marco is a minor basilica in Rome that was first built in 336 by Pope Mark. It was later restored in 792 by Pope Adrian I and rebuilt by Pope Gregory IV in 833.
St. Peter's Basilica used for Mass like any Catholic church. Of course it is most well-known for for the papal liturgies which take place there. It should be noted however, that St. Peters is NOT a cathedral.