Karlov of the Ghost Council and Extort
No, Karlov will not gain counters.

Extort will resolve before Karlov. When you gain life from Extort, Karlov will not be on the battlefield yet, and his ability only works while on the battlefield.


  702.100a Extort is a triggered ability. “Extort” means “Whenever you cast a spell, you may pay {W/B}. If you do, each opponent loses 1 life and you gain life equal to the total life lost this way.”


The order of play is:


You cast Karlov, it goes on the stack.
Because of the completed cast of Karlov, Extort triggers. It goes on the stack on top of Karlov.
Extort resolves first because it is the top-most item on the stack. You decide to pay the costs, and you gain life.
Karlov resolves and enters the battlefield. His triggered ability starts listening for life-gain events.