Is it fair for a professor to grade us on the possession of past papers?
If this was not mentioned in the syllabus (or at least announced prior to the papers being returned), this is totally inappropriate.  While it may be foolish for students not to keep their graded work (at least until the end of the term), if it was not established as a requirement before the students got their papers back, it cannot reasonably be made a retroactive requirement.

You should bring this problem to the attention of the instructor.  You do not need to reveal whether you actually have all your past papers when you bring this up, lest the instructor try to wave your concerns away if you say that you actually do have all the papers and are just complaining as a matter of principle.  I would also suggest having multiple students come to her together to make the same objection; that may make it more likely that she will relent.  However, if the instructor does not agree to make a change, you should take this up with someone higher up.  That could mean the department chair (or another departmental official, such as the director of undergraduate or graduate studies) or a student ombudsman, if one is available.