QIf this follower's Necrocharge ability triggers, and an opponent only has one follower on the field, do they only bury that follower?

AYes, they do.

QCan this follower's activated ability only be played while in hand?

AYes, that's correct.

QIf an opponent only has a Heavenly Aegis, which has Aura and "This follower can't be destroyed by abilities," on their field, can this follower's Fanfare ability cause them to bury it?

AYes. Aura prevents the card from being selected by enemy abilities, but in this case Heavenly Aegis is not being selected by an ability. Also, burying a card isn't the same as destroying it.