We were in Paris for precisely that time period last year. I was concerned that lots of things might be closed, but that wasn't the case.

Restaurants were sort of open/closed on seemingly random days, but I just made sure to reserve ahead for places I knew we wanted to go to for certain. I didn't find any that were closed for the entire period.

Some of the stores were having post-christmas sales, but the current exchange rate kept our shopping to a minimum.

Museums were pretty much open on their regular schedule, some lines were long, so I would recommend buying the Paris Museum pass that lets you skip the lines in many places.

Weather was sort of dreary, as expected.

I'd look a bit further south. Barcelona is a smashing city for a week, or you could do a combo Barcelona/Madrid.

Given that the weather will be iffy almost anywhere I'd stick with the big cities where there are lots of indoor activities and save the countryside for another trip.