QMy opponent has a Gentildonna on the field. If I super-evolve a follower, do I still give it +1/+1?

AYes, you do.

QMy opponent has a Gentildonna on the field. If I super-evolve a follower that has both an On Evolve and On Super-Evolve ability, does its On Super-Evolve ability trigger?

AYes. The On Evolve ability doesn't trigger, but the On Super-Evolve ability does.

QMy opponent has this follower on their field. If I evolve a Cursed Stone, can I play its On Evolve ability?

ANo, you can't.

QMy opponent has this follower on their field. If I play a follower with Aura, can I play its Fanfare ability?

ANo. Aura prevents a card from being selected by enemy abilities, but this follower's ability doesn't involve selection.