Lightslinger is triggered and consumed by the first attack.
The second attack will prioritize enemy champions (even if hidden in fog of war or in brush) and enemy minions with low health.
Lightslinger will still be consumed if no second target is found when Lucian fires.
Lucian can move freely while firing Lightslinger since the second attack is integrated into the first one (like Double Strike) and has special movement animations depending on the direction he is moving while firing.
Piercing Light will attempt to lead the target if it is moving but does not adjust further during cast time (enemies can dodge the laser if they change their velocity by a sufficient amount during the delay)