When are death triggers from sacrificing a creature as part of its cost put onto the stack?

ex. [[Gravecrawler]], [[Marionette Apprentice]], [[Warren Soultrader]] are on the opponent's side of the battlefield. They use Warren Soultrader's activated ability to pay 1 life and sacrifice Gravecrawler to create a treasure token. I know that I can respond to this by using a removal spell on Marionette Apprentice, but when does Marionette Apprentice's triggered ability go into the stack due to the cost of Warren Soultrader's activated ability causing a creature to die? Does removing Marionette Apprentice before the ability of Warren Soultrader's resolving mean that Marionette Apprentice won't see Gravecrawler die and I won't lose 1 life or does simply having paid the cost for Warren Soultrader's ability even though it hasn't resolved yet allow Marionette Apprentice to see that Gravecrawler died?