I understand that some dive shops are closed on New Year's day--Are all dive shops closed and will it be hard to find places to eat as well?

To my knowledge all dive shops are closed on New Years Day and if you find one that will take you, you may want to check things out pretty carefully.

I've never had any trouble getting anything to eat on New Year's Day. But then I can't remember ever trying to get something to eat before noon anyway.