QWhen evolving a follower with this follower's On Evolve ability, do I still have to pay its evolve cost?

ANo. You evolve the selected follower with this follower's ability, not its own evolve ability, so you don't pay the cost of its evolve ability.

QIf I evolve another Valiant Fencer with this follower's On Evolve ability, can I then play that Valiant Fencer's On Evolve ability and evolve yet another follower?

AYes, you can.

QIf I select a follower for this follower's On Evolve ability, can I choose not to evolve it, even if I have its evolved form in my evolve deck?

AYes, you can.

QIf I evolve Amerro, Spear Knight with this follower's On Evolve ability, do I still have to pay 3 play points?

ANo. Assuming you have an evolved Amerro, Spear Knight in your evolve deck, you can evolve it without paying 3 play points. Its On Evolve ability still triggers, so you would then select a Heroic card in your cemetery and play it for 0 play points.