If I move Akali, Deadly Weapon in, then out with her legend, does the damage stick?

It depends on what battlefield she moved to — if there were enemy units there, the damage does not stick; it's healed once the Combat Showdown ends.

Her "when I move" damage resolves first, then the Showdown opens. Activating Rogue Assassin is only legal once you're actually in a Showdown ("move a friendly unit in a showdown to base"), so by then a Combat Showdown has already started — and Combat Cleanup heals every unit in play, including units at Base that never fought (Rule 468.1.a.1). The second point of damage from the move back to Base gets wiped the same way. The exception is lethal: if the damage actually killed the unit, it dies and there's nothing to heal.

If the battlefield she moved to was empty of enemies, that's a Non-Combat Showdown, which has no Combat Cleanup — so that damage stays marked (FAQ #12550).