Is it possible to save Jigglypuff after her shield breaks? For instance, could Ganondorf jump above Jigglypuff and spike her back to the stage right after the shield breaks to keep her from dying?
Yes, it is possible for Jigglypuff not to die after having her shield broken. Upon taking damage that breaks her shield, she is stunned and flies upwards an extreme amount, most times resulting in a KO. However, the stun duration and upwards momentum is finite (several seconds), and if there is an obstacle in the way, she can survive. You can see this by going to a stage like Pac-Maze (3DS version) that has an area with a ceiling and breaking Jigglypuff's shield under that ceiling.
No invincibility frames or anything special are gained by Jigglpuff when she is stunned, and therefore she can be hit to cancel the stun and affect the momentum (assuming the attacking move is powerful enough).
As the wiki states while explaining shield breaking (bold emphasis is mine):
Players who use Jigglypuff should be especially careful about its shield breaking. Like other characters, if Jigglypuff's shield breaks, Jigglypuff will fly upward and then be stunned for a short period of time. However, Jigglypuff flies several hundred times farther upward than normal. This is more than enough to KO Jigglypuff, unless there is a ceiling or some other obstacle preventing it from flying straight up (in which case, it will helplessly keep trying to float up until several seconds have passed, the obstacle is removed, or Jigglypuff is smacked out of it).
So it is possible to hit Jigglypuff as she is launched upwards by the shield break and therefore save her, like with Ganondorf's down air as you suggested. But it would require pretty good timing, as Jigglypuff is launched upwards at a very fast rate.