There is much to do in a week. In Copenhagen there are, of course, the musea. The National Museum would be a must, if you are interested in Vikings and national treasures.
For Vikings you will find more in Jutland, e.g. the settlement Fyrkat near Hobro, the Viking cemetary Lindholm Høje near Ålborg (with a good museum connected).
A good museum for archeology is Moesgaard near Århus. They have now an exhibition about Århus at the time of the Vikings.
If you want to know more about Andersen, the first place would be the museum that is dedicated to him in Odense. I don't like this place, because it's too commercial.
About the language. How much do you want to learn? Most Danes understand English and they are more than eager to show their language skills. If you want a museum dedicated to life and letters, it's more difficult to make some recommendations. In Jutland there is a museum dedicated to Steen Steensen Blicher, who wrote some sort of peasant realism (and really good!), but it makes more sense to see these places if you are acquainted with the works of these writers.
About gay places. Pan in Copenhagen would be a good place. since it's for everybody. I wouldn't know where the queens go. You should be prepared that most of the gay places are in Copenhagen, the scene in the rest of the country is rather modest and mostly populated during the week-ends. There are some nice cafés, but you would mostly encounter mixed places.
I think your decision to see Denmark by bicycle is a good one. Do yourself a favour and have a look at the many parish churches and discover the many mural paintings.
Gentles do not reprehend.