If you bought the first printing of Gloom, it's pretty easy ... the Unhappy Homes have a blue-ish tint to the card plastic. If you bought the second printing of Gloom, you'll have to use the card lists ... you can find them by clicking on the game you want at atlas-games.com/gloom , then looking in the right column.
I'm pretty sure that icons on dead family members do count for mysteries, but I don't have my rules sheet at hand.
Again, I'd say icons on dead family members do count for Robbed by Rascals.
What are the "and the Horrid Hospital" mystery requirements? I see a Snake icon, but there's a gash in the card where the second icon should be - it basically reads "Requires [Snake], [Something] and one dead character..."
I'll double-check when I get home, but I believe it is the skull icon. The skull is the icon commonly used to represent disease - it's the story icon on galled by gangrene, suffered from sores, and the like. So the Horrid Hospital has disease and horror (the snake is actually supposed to be a creepy tentacle).
It is hard to recognize, but once you realize it's a negative image, what you've got are the two spots of the eyesockets and the teeth below.