> If we push the stick 10 cm, does it mean the stick is 10 cm shorter before the push reaches the other end?

Consider that this "stick" is not really a rigid body, but rather a large "spring", and it can compress (at atomic scale). So yes, the stick would be, from certain point of view, shorter while the push is being propagated.

> additional force to transiently deform it

Again, since it's a spring, there is no additional force needed, because the energy is conserved (aka: the spring will uncompress, releasing this "deform" energy) back.