So you have to be naked except for a goofy hat? That’s a hilarious mental image.
Actually, many people (or even majority) don’t wear it nowadays. It’s a protection against heat stroke, and if one doesn’t visit a steam room one will be fine, though it’s optionally even in a steam room, e.g. I never wore it.
Yup. The idea is that it protects your head from overheating. You wear it so that there is a pocket of air between your head and the hat, so it’s basically insulation.