If a permanent is sacrificed to pay a cost of a spell or ability, Carmen, Cruel Skymarcher's second ability will resolve before that spell or ability. Conversely, if a permanent is sacrificed during the resolution of a spell or ability, that spell or ability will finish resolving before Carmen, Cruel Skymarcher's second ability is put onto the stack.

If you sacrifice Carmen, Cruel Skymarcher, its second ability will still trigger.

If another effect causes Carmen, Cruel Skymarcher's power to be less than the mana value of the target card as its last ability tries to resolve, the target is illegal. You won't return the target card to the battlefield.