Does Akali Deadly Weapons move trigger create a window for me to play actions before the showdown?

No — Akali's "When I move" trigger doesn't create an Action window before the showdown; only Reactions can be played while her trigger is on the chain.

Here's the sequence: the move completes, a Cleanup runs, and Akali's trigger is placed on the Chain — that's a Closed State. While a chain exists, only Reaction cards/abilities are playable; Action spells require an Open State (FAQ #2235, FAQ #9306). Once her trigger chain empties, another Cleanup runs and the Showdown/Combat is staged immediately — there's no Open State in between.

Your first chance to play an Action comes after the Showdown is open and you have Focus. Since Akali's controller applied Contested, they get Focus first (Rule 345) — if you're the defender, you act once they pass or their chain empties, which is already inside the showdown, not before it.