QIf my opponent evolves an Azazel when my leader has at least 11 defense and I have this follower on my field, does my leader's defense change to 10?

AYes. Changing defense isn't the same as dealing damage.

QI have this follower on my field, and my opponent has an evolved Ginsetsu, Great Fox and a One-Tailed Fox on theirs. If they attack my leader with the One-Tailed Fox, how much damage does my leader take?

AIt takes either 0 damage or 1 damage, decided by you. When there are multiple abilities modifying damage dealt, the player receiving the damage decides which order to apply them in. In this case, depending on the order, the final damage taken can be calculated as follows:

Your leader takes 1 damage because you decide to apply Ginsetsu's ability to increase the damage dealt by 1 first, so the One-Tailed Fox will deal 2 damage before this follower's ability reduces the damage dealt by 1.

Your leader takes 0 damage because you decide to apply this follower's ability to reduce damage dealt by 1 first, so the One-Tailed Fox will deal 0 damage. Since Ginsetsu's "if a Yokai follower on your field would deal damage, it deals that much plus 1 instead" only takes effect when damage is dealt, and 0 damage does not count as dealing damage, the ability does not take effect and the damage taken by your leader remains 0.

QIf I play Jatelant, God of Prosperity's Fanfare ability targeting an enemy Shion, Immortal Aegis, do I deal 3 damage to the enemy leader?

AYes. Shion is immediately removed from the field upon being banished, so its ability is not in effect when the enemy leader takes damage.

QMy opponent has a copy of this follower on the field with 4 or less defense. If I play Dance of Death targeting it, do I deal 2 damage to the enemy leader?

ANo. Although Shion is left with 0 defense, it is not destroyed until after Dance of Death finishes resolving, so its ability remains in effect when the enemy leader takes damage.

QIf my opponent plays Dead to Rights targeting this follower, does it lose 2 attack and defense?

AYes. Losing defense isn't the same as taking damage.

QIf I play Jeanne, Despair's Maiden's Fanfare ability while this follower is on my field, does my leader lose 2 defense?

AYes. Losing defense isn't the same as taking damage.

QCan I be made to bury this follower by an ability such as Yuzuki, Righteous Demon's Fanfare?

AYes. Burying a follower isn't the same as destroying it.