Jacks overconfidence was his weakness, and omi exploited it and operated outside his own norms, and jacks misconceptions of omi, thus ensuring victory for the good guys
I believe that the killer would get the Shen Gong Wu. For showdowns that require you to find the Wu/touch it; I would guess it's not relevant because the requirement to win it is someone touching it. Also; if they're dead then they can't complete any requirements needed for the showdown, eg win the boat race, kungfu moves (Omi vs Chase) or the video game showdown.