Does Akali Deadly Weapon Ability damage happen before Attackers/Defenders get assigned?

Yes — Akali's move-trigger damage always resolves before attackers/defenders assign combat damage.

Her "When I move" trigger goes on the Chain immediately when she moves. Combat doesn't stage until a Cleanup with an empty Chain (Rule 455/460), so in the normal case (Standard Move in Main Phase) her 1–2 damage resolves before the Showdown even opens — attackers/defenders haven't been designated yet, let alone reached the Combat Damage Step (FAQ #11658).

Practical consequence: if that damage kills the only enemy unit there, no Combat ever happens — the Showdown becomes a Non-Combat Showdown and Akali conquers the battlefield without any combat damage step. Even if she's moved mid-Showdown (via a Reaction like Flash), her trigger resolves while the Showdown is open, still well before damage assignment.