If akali moves to a BF. does 1dmg. then uses the legend to move back to base doing another damage, if the unit it hit did not die, does it heal?

Yes — the damage heals. Once Akali moves onto a battlefield with enemy units, a Combat Showdown is initiated, and Combat Cleanup heals every unit in play when it ends (Rule 461.1.a.1). So after the legend pulls her back and the second point of damage resolves, cleanup returns the survivor to full health — unless the damage was actually lethal, in which case it dies instead.

This differs from the Flash trick: a reaction-speed Flash back to base means the move never becomes contested-with-enemies, so no Combat Showdown ever opens and no healing occurs (FAQ #11658). The legend route can't dodge that, because a Combat Showdown has already started by the time you can activate it.

Note this only applies if the battlefield she moved to held enemy units. Moving into an empty battlefield is a Non-Combat Showdown, which has no Combat Cleanup and therefore no healing.