QIf this follower's second Fanfare ability triggers, and I have no copies of Whitefrost Whisper in my deck, do I still have to search my deck?

AYes. After looking through your deck, you would shuffle it without adding a Whitefrost Whisper to your EX area.

QI select an enemy follower with this follower's first Fanfare ability, and then steal that enemy follower with an Angel of Chaos's activated ability. If it's engaged at the start of my next turn, does "It doesn't refresh during its controller's next start phase" apply to me?

ANo. It applies to the controller at the time the ability was played. In this case, because this follower's Fanfare ability was played before you became the stolen follower's controller, it refreshes as normal during your next start phase.

QIf I select an engaged enemy follower with this follower's first Fanfare ability, will it refresh at the start of its controller's next turn?

ANo. Although the Fanfare ability's "engage it" effect will not apply to a follower that is already engaged, the "It doesn't refresh during its controller's next start phase" will still apply.