So, if as the final action of a game, my last character on its last click attacks a Mystic, what would happen? Would my character get damaged and if so, who'd win?

You would take Mystics damage if you dealt damage with an attack to that opposing character.

If the resulting action K.O.'s all characters on the board you would roll off dice for the win. All character's are K.O.'d simultaneously.

mostly correct except the last part.

If both of you scored the most points then it goes to dice roll off.

This happened to me. If I had built my team on 1 more point, we would have had a roll off but as I KOd his last guy and he mysticked me, I scored his entire team and he mine, I got 299 to his 298