QIf my Servant of Disdain takes ability damage during my turn that brings its defense to 0, does its ability still trigger?

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

QIf I can't select another follower on the field, can I still deal 1 damage to this follower?

ANo. If you can't select another follower on the field, you can't play this follower's Fanfare ability.

QCan I select a follower on my own field with this follower's Fanfare ability?

AYes, you can.

QWhat is ability damage?

AAbility damage is all damage except for damage dealt by two followers in combat or by a follower attacking a leader. For example, if you deal 1 damage to a follower on your field with Disdainful Rending or Disciple of Disdain, that's ability damage.