I am not sure the exact travel time from Amsterdam but I do remember the train trip from Paris was 16 hours to Denmark. You might want to consider flying into London, UK, and then catching a regional airline like Ryan Air. I don't think their prices are too bad and it would save you some time.
As for driving I visited some friends in Stockholm in May and while they have a car they hardly ever use it unless they are going to the country house. Another option you might want to consider is a Scan Rail Pass which is a fairly good buy if you are travelling a bit. I took a train (X2000.maximam speed 200 km(125 miles)) from Stockholm to Copenhagen to visit some friends and it took 5 hrs. Petrol is very expensive as I am sure you know.
Let me know if you have some specific questions about Stockholm/Copenhagen surrounding area as that is where I spent most of my time.
Have a good time regardless as I had a great time in Sweden and Denmark. Tom, London, Canada