Tam-O-shanter / bonnet
snood - a small netlike cap worn to hold the hair in place.
mortarboard
No, ice cap is a common noun. It is only a proper noun if it is the name of something specific, such as Ice Cap, Alaska or Ice Cap Products.
A "house painter" wears a visored cap (if semi-spherical, a baseball cap), except that it may have only a brim, in which case it is called a visor. The stereotypical painter's cap is round but with a flattened top, similar to that of the French Legionnaire's cap.The unstructured hat worn by an "artist" may be a beretor a flat cap similar to a driving cap.
A beret is a flat, round cap. I like the tan beret better than the red beret.
A beanie or toque/tuque (Canada) is a name for a woolen cap.
He wears a red woolen cap with soft fleece inside, which keeps his head and ears warm. The hat has a white furry trim on it.
patella
kepi
stipe
The closest answer I can find is beret. The Northern Spanish do have a ridiculously large number of woollen hats though. The Catalan version is like a gnome's hat, but they somehow make it look cool!
night cap
Flat Peak hats is a Euro name for Snap backs invented in 1954 for American Bass Ball teams
Carbonated beverages will go flat overnight if uncovered. Bottles with a cap last longer, but will also loose some of their umf.
Crown, cap, or head.
hard salary cap
Cap