1) Do not even think about importing your car.... assuming you have one. The tax on importing automobiles is just absurd....
2) Start looking for a place to live veeeeery early and be prepared to move around a lot. Copenhagen rental market is as bad as they come... most of the time you will have to rent for a few weeks and then jump to the next flat for another few weeks. There are very few permanent or long term apartments available for rent in the city. Good thing: You have a chance of moving around and seeing various parts of Copenhagen and it can be quite a lot of fun meeting and spending time in the city with your new room mates.
Is it impossible to find a place to live on your own for a year or so? No. It is possible. But hard and quite expensive or in "bad" parts of the city.
3) As nacho said... don't get sick! DK healthcare is still better than the one provided in SE. But compared to CH, very basic with long waiting times. It is a good idea to get an extra insurance which will cover you in case of serious illness (like cancer or some major surgery) and offers treatment in Germany. I have one here in SE and they are quite ok... they use a network of private hospitals and guarantee to get an appointment with 72 hours. And if nothings works, they allow treatment in Germany or Poland. Actually, not that expensive. I pay about 750kr/month. But they do not cover everything... so make sure to read ALL of the conditions.
4) As in Sweden, you are nothing without the CPR. Make sure to obtain the CPR as soon as possible. Prioritize it above anything else. If you hit a roadblock, sometimes it is better to talk to an immigration lawyer. There are some big ones like Deloitte which offer these services (at 5000kr/hr of course) but also some small individual lawyers with reasonable fees to push the right levers of the machine.