Does the A-10 Thunderbolt fly without the GAU-8?
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.

http://fireaviation.com/2014/09/25/a-10-warthog-proposed-again-as-an-air-tanker/

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.