I would not, they want business casual for interviews.
No that is not appropriate
I wouldn't wear any hats to the interview
You should not wear a hat to the interview to maintain a professional appearance. But you are allowed to wear plain black/grey toques as part of the dress code.
You can but you probably shouldnt. managers are always looking for little indications such as someone who may not follow the health care or dress codes. if you do better keep it coffee related
No, definitely not.