outdoor art on the side of a van
indoor art in the National Gallery
Some things you can do while outside is look at some of the very interesting street art or around Christiania. The Tivoli Gardens are brightly lit and beautiful during the Christmas season. You can also walk around the harbor and find the Little Mermaid statue that multiple people have tried to kill (yeah, I don't get it either).
My favorite indoor place for art was the Statens Museum for Kunst (the National Gallery), which had lots of interesting installations and many works by Danish artists.
My favorite indoor places for food were:
Spiseloppen in Christiania - this place is pretty much the definition of where you want to go when it is cold outside. It's all dark wood and dim lighting and fires and hearty food. The meat entrees are all delicious.
Fiskebar - this place is located in what was formerly a meat market. The fish and chips they serve up is perfectly crispy and the drinks are also well-made. Bonus points - lots of very interesting dive bars basically form a circumference around the restaurant so you can easily have a night on the town after dinner.
Lagkagehuset - I literally went here every single day for a croissant or a pastry. So flaky and delicious. Everything is good; they even have a branch at the airport for last-minute munchies!