In this scenario, the Stratagem Sky Shadow Sync is also in play. What exactly happens when you only have Sky Shadow (Tank) and Sky Shadow (Plane) as your last characters and both get KO'd at the same time?

If all your characters are KOed, and there is no other card text saying otherwise (like a head's deploy ability), you lose. Sky Shadow Sync doesn't include such text, so if Sky Shadow (Tank) and Sky Shadow (Plane) are your last characters and both get KO'd at the same time, you lose the game.

If I attack one part of an uncombined Sky Shadow, and a triggered ability KO's that part before flipping cards resulting in Sky Shadow combining, does the attack end? Or does the attack continue against the combined Sky Shadow?

The attack continues since "Combiner Mode" is mostly another mode like Alt or Bot mode. You just can't "flip" to combiner mode without another card telling you to. Going to combiner mode is a one-way operation. Note that Sky Shadow will be untapped because the process of combining creates an untapped character.

Can Sky Shadow Sync be used in a deck that already includes Ominus, allowing you to potentially have two Ominus card's on the battlefield at the same time?

No. Sky Shadow Sync says "When ... you have Ominus outside the game". If Ominus was on your starting team, then Ominus is inside the game and Sky Shadow Sync will not trigger.