I'm curious to know if I have a pair that I want to breed in a breeding cage, would it help if I let them out in the living area to play and mingle with the other budgies or should I just leave them alone in the cage?

Budgies do not breed well with distractions. I would say ( and others may disagree ) but I would certainly not have them out playing when they are meant to be breeding......but if this is your pair where he hasnt wing feathers, there time to wait...right ?