If you stack two objects, one atop the other and drop them as a unit, do they hold that configuration for the duration of the fall or do they separate at an ever-increasing rate and distance during the fall?

In a vacuum they would stay together as long as there is no spin.

In an atmosphere it would depend on the shape and placement of each object. A non-aerodynamic front object or overlapping second object would find the air pressure forcing the two apart.