I was there last week! re: the copenhagen card It really depends on what you do. I thought it was bit of a rip off when I was looking at buying it. The public transport is pretty cheap anyway, and most things in copenhagen itself are very walkable or covered by a £2.40 tube/bus ticket.
If you were going to do a few expensive/"big" things every day or travelling outside the main 2 zone city area then maybe it's worth it. If you're more planning on staying in copenhagen city centre itself and only doing one paid/museum activity a day and wandering the rest of the time then you'd save a fortune just buying on the door.
WRT Tivoli, there isn't a "market" there as such, it's just the main parks and shops that are there all the time, just decorated for Christmas. The 4 christmas markets I went to were in Kongens Nytorv & Nyhavn (very close together), and the Hans Christian Anderson one & one by Christianborg palace (also within 5 mins of each other). I'd try and go in the night when the lights are on, tbh other than being slightly better decorated than most uk ones they weren't anything special at all in the day.
The shop 'Magasin du Nord' is like a harrods equivalent and all decorated for christmas, and is right by Kongens Nytorv. Very good for a 'Hugge' snoop but also very busy.
The Copenhagen City Museum is free every wednesday and was an interesting way to spend a few hours, although I think the main national museum is better.
Bear in mind the public transport tickets last for 1hr15mins, so you can hop on and off as much as you want during that time - useful if you just want to pop in somewhere and then see something else. You can buy tickets at the station or get the app on your phone which I found easier. Although there aren't any barriers or checks when you get on the bus or metro DO NOT be tempted not to bother - there are a lot of ticket inspectors!
Christiania is definitely worth visiting although ideally when it's dry. While you're there go to Vor Frelsers Kirke - ornate church with a huge spiral (that you can climb) and the biggest organ I've ever seen!
There was a really pretty park by the zoo - even if you didn't want to go in there are viewing areas outside to see the llamas and elephants.
Little mermaid statue very underwhelming but if you do go you can walk through another nice park and the army barracks/old windmill about 5 minutes away (Kastellet).
Think that's it!