The Pope is elected by the College of Cardinals. Papal selection before AD 1059 did not have the conclave system as is used today. The electing of the Bishop of Rome has developed over the centuries. You can read about older ways that was used to appoint a pope here: newadvent.org/cathen/12270a.htm
Theoretically yes; however, there is nobody to appoint him. Probably a pope could designate his own successor (a form of appointment), although thankfully this has never happened.
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