My wife was a math prof / teacher & engineer. I had a bachelors-level minor in math back when we wrote our proofs on papyrus using squid ink and crushed sheep shit. You totally hijacked our Friday night and destroyed a very large stack of Post-its. Thank you. You bastard! 
Our tentative but unproven conclusion in “no”. For the same reasons you can’t succeed in folding a scalene triangle in half, but generalized up one vertex.
Our collective argument rests on angles. A convex quadrilateral has 360 degrees of internal angle. In essence, your folds result in points meeting in the interior that also must meet the 360 degree criteria. And must either meet in all 4 together (picture folding in the corners of a square along creases connecting the side midpoints), or in two sets of two (picture folding a 4x4 square into a 2x1 rectangle and the generated angles and half-angles at the creases). Or meeting along a pair of sides.
For each of the above approaches given a scalene trapezoid / trapezium it’s trivial to construct a counterexample. It’s easier the more pathological your quadrilateral’s sides. An almost square is arguable, but once the sides are of length 1, 10, 100, 1000 it’s real obvious that won’t work.
FYI, the wiki on trapezoid was interesting in that it exposed a total difference in my wife’s and my’s notion of the terminology for the less-regular quadrilaterals: trapezoids, trapeziums, etc. Her and my texts had taken different sides in the terminology wars and it took us awhile to notice we were talking past one another. Gee thanks. For your entertainment: