Does Furious Finish work with both hits of Cleaving Smash?

Cleaving Smash (from Weapon Trick – Two-Handed Weapon):

When you use Cleave, you can add the additional damage from Vital Strike to both your initial and your secondary attacks. If you also have the Greater Vital Strike feat, you can instead add the damage from Improved Vital Strike to both your initial and your secondary attacks.

Furious Finish:

While raging, when you use the Vital Strike feat, you can choose not to roll your damage dice and instead deal damage equal to the maximum roll possible on those damage dice. If you do, your rage immediately ends, and you are fatigued (even if you would not normally be).

Does Furious Finish apply to both hits of Cleaving Smash, or does rage suddenly end mid-swing?

Or are they not compatible at all since Cleaving Smash doesn’t directly state that you’re actually using the Vital Strike "feat", only that you’re getting all the benefits of it?

Can Improved Cleaving Finish proc on itself?

Improved Cleaving Finish

If you make a melee attack, and your target drops to 0 or fewer hit points as a result of your attack, you can make another melee attack using your highest base attack bonus against another opponent within reach.

The way I understand it, Cleaving Finish is a passive proc. So, if I make a melee attack, kill Goblin A, I may immediately attack Goblin B. If this attack kills Goblin B, may I attack Goblin C? (D, E, F … )

(Note that Improved Cleaving Finish explicitly states that you can use Cleaving Finish any number of times per round)