Legolas, Master Archer's triggered abilities resolve before the spells that caused them to trigger. They resolve even if those spells are countered.

Legolas, Master Archer's last ability triggers when you cast a spell that has multiple targets as long as at least one of those targets is a creature you don't control. It doesn't trigger multiple times if you cast a spell that targets a creature you don't control multiple times or that targets multiple creatures you don't control.

If you cast a spell that targets Legolas, Master Archer and at least one creature you don't control, both of Legolas, Master Archer's abilities will trigger. You choose the order in which those triggers go on the stack.