Betrayal at the house on the hill arsenal room and puzzlebox
I think that none of the three interpretations you present are likely to be the intended one. 


"The text on the room tile applies only to the item you have to draw when you discover the room": This seems contradicted by the general "an" in "When you draw an item card on this room..."; if it referred to only the item from discovery it would probably say something more specific like "...the item card...".
"When you open the puzzle box you get 4 cards and gets to keep N of them": This is stretching the "choose 1 to keep. Discard the other" rather far - that language really only makes sense if the choice is between two cards.


Instead, the interpretation I'd favor is:


"Draw 2 cards" from the puzzle box means draw a card and then draw a second card, so each of those draws would get replaced by the arsenal effect and you'd end up drawing 2 and keeping 1, then drawing 2 again and keeping 1 again.


There are also two other interpretations I think are somewhat supported by the text:


"Draw 2 cards" means draw a card and at the same time draw a second card, so with the Arsenal effect that becomes drawing two pairs of cards at the same time and keeping one card from each. (This is almost like your draw 4 keep 2 option, except you have slightly less freedom in which cards to keep.)
Just draw two (with no bonus from Arsenal Room) because Arsenal Room only helps when you draw "an" item card and instead you drew two item cards. (This is the same result as your first interpretation, but for a different reason.)


The way I chose my top interpretation of these 3: Whenever rules are ambiguous, as they are all too often in Betrayal, I try to fall back on the most similar rule from MtG since they've thought of pretty much everything. In this case, rule 120.2 says


  Cards may only be drawn one at a time. If a player is instructed to draw multiple cards, that player performs that many individual card draws.