I actually first thought of Storyteller and then realized Poor House did the same. So, when I play Poor House and get to 6$, I draw only once, not twice, right? What happens when I play cards that iteratively give +1$? Vault and Storeroom for instance give "+1$ per card discarded". What about Diadem? Is it once per card?
The intention was for this to draw after each card is played that puts your total coin at $4 or more. As worded though, it would draw each time your total is increased above that amount, so cards like Vault would be quite good. Big oversight! This newest version should address this.
So now hopefully the draw card check is pretty simple and robust.
(1) Play a card.
(2) Did my total coin change?
(3) If yes and my new total is $4 or more,
+1 Card.
Applying this to your questions:
-Poor House will only get
+1 Card if upon it's resolution (1) you have above $4 coin and (2) it increased your total $. So you could have $4 prior to playing Poor House, but if it nets you $0 or less, you will not get
+1 Card.
-Cards that incrementally increase your total $ only get
+1 Card if after you resolve them your total coin is at $4 or more.
Other things worth noting:
-This interacts with all cards - Action, Treasure, and Night cards.
-Treasure cards drawn during your Buy phase can then be played to draw more cards, which can then possibly be played to draw more cards.
Thanks for catching yet another big oversight on my part!