I think leaving out the article means during one of the classes in general. Saying "the class" would more likely refer to the current class period or perhaps the one immediately proceeding.
"He's been asking many questions during class" - asking questions throughout the school year
"He's been asking many questions during the class" - asking questions this class period, although it could also mean asking questions in this particular course
"Do you want to hang out after class?" - could be asking to hang out after this class period or after a future class - although the future class case would be more likely "Do you want to hang out after class sometime?"
"Do you want to hang out after the class?" - means after this class period
*** Not a teacher ***
Student or Learner