I asked my Danish girlfriend this a while ago, and, if I remember her reply correctly, 'time' is class, as in lecture ('I'm going to class now'), and 'klasse' is class, as in subject ('which classes do you take?'). I think it's partly because classes (lectures) in Denmark are invariably an hour (en time) long.
Actually a class is usually 45 mins in Danish schools (Or at least it was before they changed the system this year. Now I don't know)
Time refers to class as in "I'm going to class" Fag refers to subject Klasse refers to grade as in "I'm in the 9th grade" "Jeg går i 9. klasse", as well as classroom Karakter is grade Forelæsning is lecture
Hope I could clear it up a bit :)
Klasse also means the classmates, as in 'klassefest'. In school, from børnehaveklasse through 3. G (HS seniors) kids stay with the members of their class, ususally numbered, as in 2b, which means 2nd grade, or 2nd year Gymnasium, the b class. Often they will stay in the b class all through school, and get to know their classmates very well. In gymnasium (at least 20 years ago) a,b,c.... are Sproglige klasser, and ...xyz Matematiske.