We had a wonderful week in Copenhagen: Beautiful, charming and quiet apartment, close to city and bus stops, little supermarket and great Tapas Bar La Esquina just across the street. In the evening it is easy to reach by foot or bus the Opera (standing tickets are quite affordable), Mojo Blues Bar (great music!) or Bastard Café/Huset House (board games + Jazz).
Due to the interesting weather (snow, sun, little rain) we didn't explore the surroundings by bike, but would have loved to do so - maybe next time.
Thank you for making us feel welcome and for the good suggestions, Maj-Britt and Michael!