You discard, put counters, and draw all while Nantuko Cultivator's ability is resolving. Nothing can happen between the three, and no player may choose to take actions.

If an effect modifies how many +1/+1 counters you put on Nantuko Cultivator, the number of cards you draw isn't modified. Likewise, if an effect modifies how many cards you draw, the number of +1/+1 counters you put on Nantuko Cultivator won't be modified.