Is there a series of rigid transformations that could map TriangleKLP to TriangleQNM? If so, which transformations?

No, TriangleKLP and TriangleQNM are congruent but TriangleKLP cannot be mapped to TriangleQNM using a series rigid transformations.
No, TriangleKLP and TriangleQNM are not congruent.
Yes, TriangleKLP can be reflected across the line containing KP and then translated so that P is mapped to M.
Yes, TriangleKLP can be rotated about P and then translated so that L is mapped to N.