Germany in January

Frankfurt is on one of the warmer areas of the country and of late has had relatively "warm" winters, but there's no guarantee you won't freeze your fanny off. I've seen snow there in April, and routinely in January - sometimes a LOT of it, with sub-freezing temps. I was in the area a couple of years ago in January and had some winter sunshine with temps near 50 F. It just depends on your luck whether you'll be able to enjoy the outdoors. You can generally visit the same Frankfurt tourist sites in winter as in summer.

Completely closure is not likely, save for summer excursions like leisure boar trip in the Rhine, certain parks and hikes etc. More likely are reduced opening hours. As for the weather: it's not unbearable, but be prepared like you would for a trip to - say - Pittsburgh.