Does akali when she is empowered and gets in and out twice this turn and hits my units does the damage stays on my units or when there isn’t any showdown to begin with my units heal?

No — without a Combat Showdown nothing heals mid-turn: the damage stays marked on your units, but it only matters if it's lethal, because all marked damage is healed at end of turn anyway (CR 317.2.b).

The heal you're thinking of only happens in Combat Cleanup, which requires an actual Combat Showdown — i.e. Akali arriving at a battlefield where your units are and surviving until the Showdown opens (CR 468.1.a.1). If no Combat Showdown ever opened (empty battlefield = Non-Combat Showdown, or she was bounced back at Reaction speed while her move trigger was still on the Chain), there's no heal, so both 1- or 2-damage pings stay marked on your unit for the rest of the turn.

Worth flagging for the "in and out twice" part: the Rogue Assassin legend can only be activated while a showdown is ongoing, so any pull-back with it means a Combat Showdown already opened there — and when that combat resolves, Combat Cleanup heals every unit in play, including ones at Base, wiping all her non-lethal damage. Each re-entry is a fresh combat with its own Cleanup, so per-entry damage evaporates unless it kills.