Portsmouth usually puts on a fun First Night (12/31), with fireworks and such. Consider sticking around for that.
Regarding that week... I don't believe places close. You should be fine.
Regarding places to eat... there's a fun hot pot place on Penhollow st (Pink Bamboo). Jumping Jays is my favorite for seafood in town. I just had a recommendation that Massimo's was the best italian meal an acquaintance ever had. If you have a car (or don't mind ubering), some good fried seafood is at Bob's Clam Hut in kittery (right near a bunch of outlet shopping).