Top hat and tails
Yes, "Top Hat" is a proper noun when referring to the formal hat worn as part of a formal outfit.
A bucket hat
boaters
your ugly face
The name of the hat with a flat top worn at graduation ceremonies is called a mortarboard.
Tuxedos may be worn with an appropriate hat. This may be a bowler's hat or a top hat, among others.
It is unknown who invented the top hat. It is thought to be descended from the sugarloaf hat. George Dunnage created the first silk top hat in 1793. It was only in 1797 that the top hat made its first appearance, when it was worn by English haberdasher James Heatherington.
boater
It's a boater.
Mortarboard
No, it's an article of clothing worn on top of the head.
type of hat worn by witches