Owning a dog in Copenhagen - where to live?

Hello there, I am about to move to CPH with my dog. My dog is used to run off leash in nature, where we go for 1 hour hikes at least 4 days a week. So my question is: are there places in CPH or just outside, where dogs can do that?

Also, I have seen that there's dog walkers that take dogs for longer walks in nature, which is something that I will also look into, as my partner is not coming with me, at least for now. These are usually tied to Norrebro/Osterbro etc. - so would that be a good place to look for flats? There seem to be dog areas in the bigger parks, that would at least suffice for those days I don't have time for a proper hike.

And coming from a place where there's an inofficial rule that dogs are allowed off leash in the early mornings in parks - does a place like this exist in CPH?