if a Haxorus with Mold Breaker uses False Swipe on a Shedinja holding a Focus Sash, assuming that False Swipe would hit a Ghost Type, would Shedinja's Focus Sash be broken, or would the game say that Shedinja wasn't damaged because False Swipe technically didn't reduce Shedinja's HP and also did not activate Sash since FS keeps a Pokemon from dropping below 1 HP??
False Swipe would fail. Nothing would activate.
Don't know how this scenario would come about, but the Shednija's Sash would not be affected.