QI have a copy of this follower on my field that's been given Ward by something like Death's Breath or Guardian Sun. An enemy follower attacks my leader. If I engage this follower with Quick timing to pay for its activated ability, does the enemy follower's attack target change to this follower?

ANo. An attack target has already been selected, so the attack goes through.

QWhen my opponent has a Lyrial, Archer Throne with 5 or less defense on their field, why does playing Dance of Death not deal damage to the enemy leader, while playing Fiery Embrace with Spellchain (10) does?

AIt has to do with the timing at which the enemy Lyrial is destroyed, or in other words, the duration for which its ability remains valid.

First, let's look at Dance of Death. When a follower's defense becomes 0 or less, it gets destroyed at the next Confirmation Timing, which occurs after the current ability—Dance of Death, in this case—is resolved. Therefore, when Dance of Death deals 2 damage to the enemy leader, Lyrial has 0 or less defense, but it's still on your opponent's field, so its ability is still valid, preventing its leader from taking ability damage.

Now let's look at Fiery Embrace with Spellchain (10). Effects indicated by an ability are applied in the written order, so the instruction "destroy it (the enemy follower)" is executed first, and Lyrial is immediately destroyed. By the time the instruction "deal 3 damage to that follower's leader" is executed, Lyrial is no longer on the field, so its ability isn't valid, and the enemy leader can take ability damage.

QIf I deal 1 damage to my leader with an ability while this follower is on my field, my leader doesn't take any damage, but does Sanguine still become active for me? Does it count toward the number of times my leader has lost defense this turn?

ANo, and no. Sanguine only becomes active for you if your leader loses defense during your turn.

QIf my opponent's leader has at least 11 defense and they have Lyrial, Archer Throne on their field when I evolve an Azazel on my field, can I change my opponent's leader's defense to 10?

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

QIf I pay the cost of Demonium, Punk Devil's activated ability while this follower is on my field, do I still give my leader -2 defense?

AYes. Giving -2 defense isn't the same as dealing damage.

QIf I evolve a Humpty Dumpty on my field when my opponent has a Lyrial, Archer Throne with 5 or less defense on their field, can I deal 3 damage to their leader?

ANo. Humpty Dumpty's On Evolve ability deals 5 damage to Lyrial, bringing its defense to 0 or less, but it's not destroyed until Humpty Dumpty's ability is fully resolved, so Lyrial's ability is still valid when you would've dealt 3 damage to its leader.

QWhat is ability damage?

AAbility damage is all damage except for damage dealt by two followers in combat or by a follower attacking a leader. For example, damage from an evolved Izudia, Omen of Unkilling's activated ability, "Deal 6 damage to each enemy leader with at least 10 defense," counts as ability damage.

QIf I play Fiery Embrace with Spellchain (10) and select an enemy Lyrial, Archer Throne, can I deal 3 damage to its leader?

AYes. By the time you resolve "Deal 3 damage to that follower's leader," Lyrial is already destroyed and no longer on the field, so its ability is no longer valid.

QIf I play Dance of Death and select an enemy Lyrial, Archer Throne with 5 or less defense, can I deal 2 damage to its leader?

ANo. Dance of Death deals 5 damage to Lyrial, bringing it to 0 or less defense, but it isn't destroyed until after Dance of Death is fully resolved, so Lyrial's ability is still valid when you would've dealt 2 damage to its leader.