Health code generally doesn't allow this. The purpose of the hat is to prevent hair from getting in customer food. If you wore a hairnet under it then that would pas health code, it would then just be a question of if it was in dress code.

It depends on the state laws where you work. I’d ask your manager.

No, my boyfriend got in trouble for this.

I have a coworker who wears bandanas. I would check with your manager

Only if it's required by your religion.