"The third card of your deck" refers to the third card from the top of your deck, of course.

Putting 1 Air Mana under your Crow is one of the condition to Draw, if you have no Air Mana to put under it, you do not draw a card.

In general, if you have drawn more cards than required, you can still apply the effect of your Crow's first ability. (So yes, in the first turn with your crow, it is indeed considered that you drew 0 cards this turn at some point).