QIf a follower with a Fanfare ability is put onto my field while this follower is on my field, do I resolve its Fanfare ability before this follower's "Whenever a follower is put onto your field" ability?

AYou may resolve them in any order. When a follower is put onto the field, its Fanfare abilities and any "Whenever a follower is put onto the field" abilities trigger and become pending at the same time. Therefore, you can resolve those abilities in any order.

QIf I evolve a follower with this follower's automatic 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.

QWhile this follower is on the field, can I choose not to evolve a follower put onto my field?

AYes. If you don't reveal a follower's evolved form from your evolve deck, that follower won't evolve.