does shuriken flip kill before showndown is staged from its move

Yes — the kill happens before the Showdown from the move is staged.

Shuriken Flip resolves as one Chain item: "Deal 2," then the move applies Contested. The damage only marks lethal damage; the actual kill occurs in the Cleanup that follows resolution — Cleanup step 3 kills lethal-damaged units (Rule 323.4–323.5), while Showdown/Combat staging doesn't happen until steps 6–7 (Rules 323.8–323.9). Since the enemy unit is already in the trash by then, no opposing units are present, so no Combat is staged — if the battlefield is still Contested, a Non-Combat Showdown opens instead (Rule 344.2; see FAQ #11630 and #11647 for the same cleanup ordering).

Practical result: your moved unit wins a Non-Combat Showdown, establishes Control, and Conquers. If the target survives (3+ Might, or damage prevented), then opposing units are present at staging and it opens as a Combat Showdown instead.