this is almost as if the user is not sure the new state has occurred yet.. did the doors open ("ka dou ka") or not yet..
Literally, it means "while it is not certain whether the door opened or not," so you cannot tell if the door opened or not. It depends upon the text that follows, but normally, it would mean that the door did open.
#5243 授業（終了の）ベルが鳴るかどうかのうちに is better. I would say from the English... 授業終了のベルが鳴るか鳴らないかのうちに、学生たちは教室から飛び出しました。 This does not mean students flew out, by the way.
actually, if you said, 'the students flew out' that would be acceptable english. it is understood that you mean that they moved quickly, not that they flew like birds.
are we sure this is a real and grammatically correct phrase? Google doesn't have many hits for it. Maybe a note about how it is a replacement for another phrase...