QIf my 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 On Evolve ability cause them to bury it?

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

QMy opponent has two 5-attack followers on their field and nothing else. If I play this follower's On Evolve ability, do they choose which one to bury?

AYes, they do.