Perhaps if you show me some of your drawings, I can see where my description may have been ambiguous. That is one problem about communicating when one person can see the diagram and the other cannot.
With respect to the rigidity question, I think that the best way to look at the problem is to start with the full convex polyhedron, say the cube, and see how many faces you can remove before it becomes flexible. I know that if you remove two opposite faces, you get a quadrilateral cylinder that is certainly flexible. But I suspect that if you remove even one face, you will be able to flex the resulting figure, once you introduce various diagonals into the square faces You could prepare for this by scoring all square faces with both diagonals, then remove one face and see what happens as you begin to pinch two of the boundary vertices toward each other.