QIf I play Apostle of Craving from my EX area, does this follower's "Whenever you play a Neutral card" ability trigger?

AYes. Playing a Neutral card (tokens included) from any zone, whether it be your hand, EX area, or cemetery, triggers this follower's ability. However, note that evolving a follower doesn't count as playing a card.

QIf I play this follower, does its own "Whenever you play a Neutral card" ability trigger?

ANo. When you play this follower, it's not on your field yet, so its abilities aren't valid.

QCan I choose the order in which I resolve this follower's two Fanfare abilities?

AYes, you can.

QIf I put Lucifer onto my field with Death's Breath while this follower is on my field, does this follower's "Whenever you play a Neutral card" ability trigger?

ANo. "Putting" a card onto your field with an ability isn't the same as playing it.

QIf I don't have a Craving's Splendor in my deck, can I still search for an Apostle of Craving and put it into my EX area with this follower's Fanfare ability?

AYes. Note that even if you do have a Craving's Splendor in your deck, you can choose to not find it.

QIf I play a Neutral card and deal 1 damage to the enemy leader with this follower's ability, does my leader get +1 defense from Drain?

ANo. Drain only triggers when this follower deals attack damage, not ability damage.