if akali moves to a BF and does 1dmg. then leaves and does 1dmg. but the unit at the BF does not die. does it get healed since the showdown/combat ended?

No — the unit doesn't heal, as long as the pull-out happened before an actual Combat Showdown opened.

This is the classic Akali pattern: you move her in, hold her "When I move" trigger on the Chain, and reaction-speed move her back out (e.g. Flash). Because no item-free Cleanup ever occurred with you still present, the Combat Showdown never opens — Contested just gets removed in Cleanup (CR 323.10), so there's no Combat Cleanup and therefore no healing. Both 1-damage triggers resolve and the damage sticks (FAQ 11658).

The distinction that matters: healing only happens during Combat Cleanup after a Combat Showdown (CR 468.1.a.1) or at the end of the turn (Rule 317.2.b). If a Combat Showdown did open — say you used the Rogue Assassin legend at Action speed while a combat was already running — then Combat Cleanup would heal the surviving unit back to full unless the damage was lethal.