Citation in examination
If the case is fairly stated then the professor is not being reasonable. Students shouldn't have to cite what they study prior to the exam. In fact, in order to provide a proper citation the student might actually have to return to the site to get the proper citation, breaking the rule in order to uphold it - Catch 22. 

However, since "innocent until proven guilty" may not be the operative standard (sadly) it may be difficult for the student to make the case. But it is worth making and should be made forcefully. Appeal to higher authority if necessary. 

I will leave your last two questions unanswered as they are treated elsewhere on this site and don't really affect the outcome for the student, which I treat as paramount here.