I see many resources on how to prevent a Double Leg takedown from even occurring initially, including Sprawl technique, or knee to head etc. However, how do I defend against a double leg takedown when an opponent has me on the shoulders and in the middle of the air? Do I just prepare for better luck and go with a Judo fall, or should I attempt to elbow opponent in mid air?
In a self-defense scenario, your primary objective is to protect yourself. Tuck your head forward to avoid it impacting the ground. If possible, pull your shoulders forward to activate the muscles in your back to ensure that the meat of your back, and not your spine, hits the ground, and try to turn slightly to one side so that you don't land flat on your back and get winded. I believe (I don't have statistics) that most self-defense double-legs do not involve much of a lifting off the ground, focusing more on taking the other person down, so there will really not be time to try to secure strikes on the other person before impact.
On the bright side, in a self-defense scenario, you're likely to be dealing with someone who either has learned double-leg secondhand from watching the UFC, or comes from an amateur wrestling background, so once you've ensured you do not sustain serious injury by tucking your head and protecting your spine, you're more likely to have an opening to start pummeling at their head after the impact, and if they do try to lift you, especially for an untrained individual, there's a decent opportunity to try to throw them off by twisting and shifting your weight during the lift.
Disclaimer: this comes from someone who doesn't train. At all.
Self defense: gauge their eyes with your thumbs. One at a time. That's got to hurt. A strike beforehand would aid their cooperation in the endevour.
Sparring: cling on. If you hug the person it's difficult for them to throw you to the ground. And even if they do it gives you wrestling options.