Hey, just checked.
It properly reduces cooldown.
>"deals damage to an enemy" means any damage? Including from debuffs like fire, curses, etc?
DoTs don't count. Status effects like burn, shock, and bleed don't "hit," but they also ignore evasion.
As for other effects, it depends, but they usually inherit the damage source (which is a unit).
>Does this state stacks? Like Proxy set?
No, "Moon's Favour" buff doesn't stack.
Get well soon! Take care.