Also would a spring under tension greater than the force of gravity pulling the spring downward shrink in both directions until it has depleted it's stored energy while in free fall?
As long as the total weight does not exceed the elastic limit, then yes.
Why? For any given length element, the weight that it is supporting doubles, so the change in effective length of that segment also doubles as per Hooke's law.
The limit is when the most strained element passes it's elastic limit. That will be the topmost bit which is holding the weight of the whole thing.