QDo I evolve this follower with its Fanfare ability if it is put onto the field by an ability during an opponent's turn?

AYes, you do.

QIf I summon this follower from my deck with Aenea, Amethyst Rebel's Fanfare ability, can I play its own Fanfare ability to evolve it?

AYes. Since it was put onto the field by an ability, it would evolve.

QIf I evolve this follower with its Fanfare ability, can I evolve another follower using its evolve ability in the same turn?

AYes. Although you can only activate one evolve ability per turn, other abilities that evolve followers do not count towards this limit.