QMy opponent has a follower on the field that was given Drive by its Ride ability. If that follower becomes Boxed, does it regain Rush and Single Drive when it is no longer Boxed?

AYes, that's correct. Boxed followers will regain their abilities once they are no longer Boxed.

QMy opponent has a follower on the field that was given Ward by Death's Breath. If that follower becomes Boxed, does it regain Ward when it is no longer Boxed?

AYes, that's correct. Boxed followers will regain their abilities once they are no longer Boxed.

QIf I select Iceschillendrig, Gilded Autocrat with Nahtnaught, Cursed Queen's activated ability, does Iceschillendrig, Gilded Autocrat's automatic ability trigger?

AYes, it does. You would then resolve it at the next confirmation timing.

QIf I select an engaged enemy follower with this follower's activated ability, will it still become Boxed until the end of its controller's next turn?

AYes, it will. Although the activated ability's "engage it" effect will not apply to a follower that is already engaged, the following effects will still apply.

QIf a follower given the Armed trait by Draconic Weapon's effect becomes Boxed, does it lose the Armed trait?

ANo, it doesn't.

QCan a Boxed follower be given Ward by an ability such as Guardian Sun's Fanfare?

AYes, it can.

QIf a Boxed follower with a "When this card is returned to hand from your field" ability is returned to its owner's hand from the field, does the ability trigger?

ANo, it doesn't.

QIf a Boxed follower with a Last Words ability is put into the cemetery, does the ability trigger?

ANo, it doesn't.