With the introduction of the Gold Rush expansion, there is no FAQ on how to handle Lucky Duke/Horse Shoes with Rum.
Lucky Duke says each time he "draws!" flip the top two cards and chose one. Horse Shoes says whenever you "draw!", you reveal 1 additional card and choose the result. Rum says "Draw!" 4 cards: you regain 1 life point for each different Suit revealed.
So how do these interact? It's not clear if Rum means to draw 4 cards, or draw 4 times. My play group has ruled that it is intended to draw 1 card 4 times, which would mean Lucky Duke and Horse Shoes would take full advantage of it. But that may not be the case; it's hard to say since Lucky Duke was created before there were any cards that had a draw of multiple cards.
Here are the different ways I can see this working.
1) Based on the wording of Lucky Duke he only draws two cards for Rum and would draw a 3rd card if he had Horse Shoes, putting his use of Rum at a disadvantage compared to all other characters. This would also mean all other characters would draw 5 cards from Rum and they would chose the 4 that they would want.
2) Based on the assumption that Rum is intended to draw 1 card 4 times, Lucky Duke would get 4 draws of 2 cards, and chose one after each two card draw to determine how much life he gets back. If he has Horse Shoes he would draw 3 each time and any other character would draw 2 each time.
From my searches I have seen people arguing for both sides. Personally I'm thinking the second choice is right since it seems that Lucky Duke should have an advantage when it comes to draws.