QIf I play an Airbound Barrage targeting this follower on my field, does its "Once per turn, when you play a spell" ability trigger?

AYes, it does.

QMy opponent has a Rulenye, Omen of Silence on the field. If I play this follower's Fanfare ability, then play an Airbound Barrage, how much does it cost?

AIt would cost 1 play point.

QIf I play a spell during my opponent's turn, does this follower's "Once per turn, when you play a spell" ability trigger?

AYes, it does.