QIf this follower is selected and destroyed by Execution, does this follower's ability trigger?

AYes. You resolve this follower's ability after Execution is resolved.

QIf this follower takes damage from an ability that doesn't involve target selection, like Angelic Barrage, does its ability trigger?

ANo, it doesn't.

QWhen playing Armor Burst, if I select this follower and an enemy Goblin, what happens to this follower?

AFirst, you resolve Armor Burst, which puts this follower into its owner's EX area. Then, you resolve this follower's ability, which triggered when you selected it with Armor Burst. The effect "give it +1 attack" refers to this follower as it was on your field when the ability triggered, so now that it's in the EX area, it doesn't gain the buff. However, you still deal 1 damage to each enemy leader.

QIf I select this follower with Draconic Weapon's ability, does this follower's ability trigger?

AYes, it does.

QIf my opponent selects this follower with a Quick spell during my turn, does this follower's ability trigger?

AYes, it does.