The pope traditionally wears four objects on his head.
It is important to note that the Vatican is a true independent state, with civil authority. The pope is the head of this state. The papal states, and the military power they contained, were quite large centuries ago (up until 1871). Hence the appropriateness of the crown.
The tall bishop's hat is called a mitre (pronounced miter) while the skull cap ('beanie') is called a zucchetto. The Papal crown or triple tiara is no longer worn by the pope since the death of Pope Paul VI.
That depends which hat you mean.
There have been 13 popes called Leo, and there appears to be no dispute about the numbering, as is the case with a few of the popes. See links.
There have been 8 popes called Urban.
Avignon, France, is sometimes referred to as the "City of Popes" because of the presence of popes and anti-popes from 1309 to 1423 during the Catholic schism.
There have been 13 popes named Innocent.
cap and gown. the hat is called a mortarboard.
It is called a mitre and is also worn by all bishops, archbishops and cardinals. It is symbolic of a bishop. And is also considered a tradition as it dates back to the medieval days.
a leperchaun's hat :P
It's called a homburg hat
Yes, they had many differences on matters of authority and doctrine. Some Popes were even excommunicated by Church Synods (or Councils) and other rival Popes were elected in their place, called Ante-Popes.
Its called a hat box
There have been 6 popes who chose the name of Paul.
According to Irish folklore, it's called a cocked hat.