If Diana Lunari is dealt lethal damage by a Shurkien Flip and an empowered Akali Deadly weapon moving into that battlefield, can I still trigger Diana Lunari's ability?

Rule references correction

No — Diana's ability won't trigger, because she's dead before the Showdown actually begins. Her trigger checks "when a showdown begins here," and by the time the Combat Showdown opens, she's no longer at that battlefield. Moving is instantaneous, and Akali's "When I move" trigger is placed on the chain and fully resolves before any showdown can start — a Combat/Showdown only begins in a Cleanup with an empty chain, and showdowns/combats are staged and opened at the very end of the cleanup (Rules 323.8–323.13, 460). So Shuriken Flip's 2 damage resolves, then Akali's move trigger adds 2 more, and Diana (3 Might) dies during the cleanup at step 3b — which runs before Step 9/10 begins the Showdown/Combat (Rule 323.5). By the time the Showdown begins she has already left the battlefield, so the "when a showdown begins here" condition can't be evaluated for her (Rule 383.2.c.2). This is different from removal after the fact: if her trigger were already on the chain, it would resolve independently of her (FAQ #10222) — but here it never gets placed at all. Note also that the Akali trigger damage is what makes the damage lethal; either way it resolves pre-showdown, and if Diana were the only opposing unit, that battlefield becomes a Non-Combat Showdown.