In Copenhagen we stayed at the First Hotel Skt Petri which is in a fantastic location, central and close to all the main attractions. It's also only a few blocks from Ørstedsparken. Dogs are meant to be on-leash in the park but I saw this loosely enforced. Also, next to the park in the median strip on Nørre Voldgade is a fenced in dog park. I know it sounds strange, but it works. Copenhagen is known for bicycles and being very cycle friendly - and as a result there is little car traffic in the centre relative to other cities.
We were only in Copenhagen for a day and a half so not enough time to really explore. I talked to some dog owners in the park, though and they said dogs are allowed into a lot of cafes but that there aren't enough off-leash places.
Also, I forgot to bring dog food and the hotel very kindly made us up a plate of steak and potatoes. All organic, of course!