QIf I play Raio, Truthful Elimination, burying a Disciple of Truth as part of its cost reduction, does Disciple of Truth's automatic ability trigger?

ANo, it doesn't. Since it is buried as part of the cost, Disciple of Truth is no longer on the field when Raio, Truthful Elimination is played.

QIf I play this follower, does it trigger its own ability since it's a Mage card?

ANo. When you play this follower, it's not yet on your field, so its ability isn't valid.