Folding is allowed, yes.

Parts can have different shapes (if you want), but must have the same area.

You cannot use compasses, but you can use an unmarked straightedge if you want to make precise creases, or to avoid ripping the paper in an untidy fashion. You are not allowed to mark the straightedge, of course.

If the procedure were carried out with infinite precision, then it would indeed produce exact fifths.