Hej! I just spent 6 months in Billund (middle of Denmark) doing an Internship at LEGO.
Cheap and good in Copenhagen is to catch a canal boat ride in Nyhavn and go out into the harbour, and you can drink on board (if the sun is shining it is glorious).
The Design Museum is pretty good but small- they had a good children’s design exhibition on in June.
I liked the Tivoli but it is expensive, however from the starflyer (swing carousel ride) you can actually see Sweden- one of the scariest things I’ve ever done.
The meat packing district is nice (but expensive) and the food market there is good.
Christiana was pretty ordinary, lots of burnt out hippies and radicals, too many rich tourists on hotel bicycles…