A: No. You cannot apply the No-Hand Bonus retroactively. It’s time of effect, “before observing the current Draw rule,” starts – and ends – before you’ve played it, so its first potential usage will necessarily be the next player’s turn.
For more info on the phases of a Fluxx turn, see Order of Events in a Fluxx Turn (also linked in the lower right sidebar).
Also see: Does Inflation apply retroactively to the No-Hand Bonus?”