I move my Akali - Deadly weapon into a battlefield solely with oponent's Akali - Silent. Do I deal damage? If I react to my Akali move trigger with flash, does it deal any damage? If instead I use Akali, Rogue Assassin ability, does it deal any damage?

No for the first two, yes for the third — it all turns on whether Akali, Silent is "in combat" (i.e. holds an Attacker/Defender designation) at the moment your move trigger's target is chosen.

1. Straight move in: no damage. Your "When I move" trigger is finalized — targets picked — the instant she arrives, and a Combat Showdown only opens during a Cleanup with an empty Chain (Rule 460), so Akali, Silent has no Defender designation yet and isn't "in combat." She's Untargetable to your ability, so you can't choose her and nothing is dealt (cf. FAQ #12466, #11816).

2. Flash in response: still no damage. Flash sends her back to base and adds a second "When I move" trigger (the battlefield she moved from), but the Chain never empties, so no Showdown ever opens and Akali, Silent never gains a designation — neither trigger can choose her. This is the opposite of the usual Akali Flash trick, which only works because a normal unit is targetable (FAQ #11658).

3. Rogue Assassin legend: yes, 1 damage (2 if Empowered) — but it's healed. You can only activate the legend once the Showdown has actually started, which means it's already a Combat Showdown and Akali, Silent is the Defender (Rule 464.2.c.3 — designations persist until combat ends, Rule 466.7.a). She is in combat, so choosable, and the pull-back trigger hits her. Combat Cleanup then heals every unit in play, so unless the damage was lethal she ends at full health (FAQ #12550).