Has the A-10 Thunderbolt ever flown without the GAU-8? Can it be done with a ballast in place of the gun? I'm not referring only to the combat role of the A-10, a plane used for training or a ferry flight would also qualify.
Near as I know, the answer to the first question is no. The aircraft requires that a tail stand be fitted to it when the gun, magazine and electromechanical drive mechanism is removed for maintenance. Operation of the A-10 without the gun would exceed the weight and balance envelope for the aircraft, making flight, as well as ground operations hazardous.
The A-10 has been proposed for use as a converted water bomber to fight forest fires but this balance problem without a counterweight installed in this region would make this very difficult.
If one was going to remove the gun for a ferry flight, it would have to be fitted with some kind of counterweight in its place, which would defeat the original purpose of removing the gun.