QIf this follower is given -1 defense during my turn, does its ability still trigger?

ANo. Being given -1 defense is different to taking damage.

QIf this follower takes damage that brings it to 0 defense or less during my turn, do I still select an enemy follower on the field and deal it 3 damage?

AYes. Taking damage triggers this follower's ability. Then, since the follower has 0 defense or less, it is destroyed. Then, you resolve the ability.