QIf there are no enemy followers on the field, or there are only followers with Aura, can I still play this follower's "Select an enemy follower on the field and engage it" activated ability?

ANo. If there are no enemy followers on the field that you can select, then you cannot play the ability or pay the cost of putting this amulet into your cemetery. Note that if there is an engaged enemy follower without Aura, you may select it and play this ability. In that case, the selected follower's state would remain unchanged.

QWhen I play this amulet's "Banish each follower on the field" activated ability, will it only banish enemy followers on the field?

ANo. All followers on all players' fields will be banished.

QCan I pay the cost for either of this amulet's activated abilities when it's engaged?

ANo. Both costs require you to engage it, so if it's already engaged, you cannot pay either cost.