Quote from Rezzahan »only if Kozilek has not been countered at that time already, which is what a smart opponent would usually do: counter Kozilek before you draw more cards and a possible answer. In other words, a smart opponent would respond to the trigger and not wait for Kozilek to be the top most spell on the stack. You can target objects on the stack that are not on top.
Quote from peteroupc »When you get priority, you can cast an instant spell even if it's from a card you just drew due to Kozilek's first ability (C.R. 117.1a). However, you can achieve what you want (that is, cast a card drawn due to Kozilek that would counter a spell that in turn would counter Kozilek) only if Kozilek didn't leave the stack before its first ability resolved (e.g., because a Cancel, which targeted Kozilek while Kozilek and that ability were still on the stack, countered Kozilek).
Quote from peteroupc »Even so, however, you can't activate Kozilek's last ability yet (even if you have a card with converted mana cost equal to that spell's in your hand) since Kozilek is still on the stack and that ability works only while Kozilek is on the battlefield (since it could work there and doesn't mention another zone where it works) (C.R. 113.6).
Quote from MadMageQc »If you mean to counter a counterspell that targets Kozilek himself, no, not if your opponent is doing things right in their own interest. The card-draw ability goes on the stack above the Kozilek spell when you cast him, and then your opponent can cast his counterspell targeting Kozilek in response to the draw trigger, and it will also resolve, countering Kozilek, before you draw cards.