No, it is not a requirement.
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.
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.
The Basilica is a monument to Saint Peter, the first Pope, who happens to be buried under it. It is the church used by the pope for important celebrations, such as canonizations and Christmas Eve midnight Mass.
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.
A church may be called a basilica because of a particular event (miracle, pilgrimage) or because there are relics in it. The name is largely honorific and the right to be called basilica is given by the Pope. A church may be called a cathedral when it is the seat of the diocese (and hence headed by a bishop), which is not the case for the basilica of the Sacré-Coeur.
In 2014 the current pope is Pope Francis.
The Pope's home church, whether it's Julius II or any other Pope before and after him is the Basilica of St. John Lateran in Rome.
His tomb was originally intended for St. Peter's Basilica but was instead placed in the church of San Pietro in Vincoli on the Esquiline in Rome after the pope's death. Julius had been the titular cardinal of this church before he was elected as pope.
Benedict is now retired and has nothing to do with the Basilica.
He celebrates Mass everyday. If he is not scheduled to say Mass at St. Peter's Basilica or St. John Lateran or at another church, he says Mass in his private chapel in the Vatican Papal Apartments.
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.