The white cap is a zucchetto. The ceremonial head covering that looks kind of like the Chess piece it inspired is a mitre, and the triple crown shown on the papal flag above the crossed keys is called a tiara.
The pope's crown is called the papal tiara. There is a link below to an article on it.
Charlemagne was not prepared for his coronation and may not have wanted to be crowned by the Pope. If the Pope had the power to crown Charlemagne king, the Pope might also have the right to remove the crown.
He insisted on the presence of the Pope, but the Pope did not crown him. Napoleon put the crown on his own head.
It is called the Papal Tiara or Papal Crown, sometimes called the Triple Tiara or Triple Crown.
The Pope
charlamagne/chareles the great
There have been 6 popes named Paul so you need to be specific. However, the papal triple tiara (crown) has not been used by pontiffs since Pope Paul VI and there is no reference to anything being inscribed on his crown.
800 CE
Charlemagne in 800 CE
So that he would not appear subservient to the Pope.
The Black Pope's birth name is Shelly Pope.
Napoleon crowned himself Emperor. This was already planned, and Pope Pius VII agreed.