Thanks for your questions “What are some of the best vegetarian and vegan restaurants in Copenhagen?”
I think this question is very important for all like yours. Here I’m trying to give your answers in brief.

Firefly Garden on Fredericksborggade is the one completely ‘high-end’ vegan restaurant in town. It is a bit expensive, but quite good.
Morgenstedt in Christiania is cheap and funky and good, all vegetarian.
Green Sushi (multiple locations) has a great innovative vegan selection, not just the usual suspects.
Café N on Blågardsgade is all veg. Nice & cozy.
Von Fressen in Vesterbro has good veg lunch & dinner options.
Mahalle in Nørrebro serves a great Lebanese veg plate.
Spiseloppen in Christiania for a good dinner. They also serve meat, so it is a nice place to go with a group of mixed people.

UPDATED: I just had an incredible meal at Verandah, the high-end Indian restaurant. The vegetarian prix-fixe is one of the best meals I have had in a long time!

Those are my favorites!

Thanks for your questions. If you find this answer is useful for you then share with your friends on facebook, twitter and google plus. If you do not get your answer clearly then ask me again below or comment on my answers.
If you have any questions, please feel free to ask me here.