I would probably say "Luckily, all the students were in class at the time." because "in class" indicates better that they were physically in a different location than "having class" or "having classes" does. "Having class(es)" would be understood and not be wrong, with the main difference being that "having classes" is specific that there was more than one class going on at the time of the accident while "having class" is general that they were in school.

Hmm... This is sort of difficult. To me, "having classes" sounds better, although I think the first option still sort of works. I would say either "All the students were having classes" or "All the students were in class" although having class probably does make sense too... I'm a native but I am still a bit stumped! Sorry I can't help more.