Lost City, less specific:spoiler:If you read the book, at the end of it you will get a clue for the Lost City.
Specific:spoiler:Grab the potion, use it, shoot the rug, turn around.
RE: getting into the area below the Gallery: spoiler:Do you mean the little square of wood in floor of the inventor's shop with all the boxes over it? I thought it might be a secret door or something, but couldn't find a switch.
EDIT: spoiler:Found the switch, just ended up in the vault again. Found another key in there, though, and got the Horn Of Quintus. Yay. That makes a total of four boxes I was able to open. Duh. I swear I tried picking those doors below the gallery before and it didn't work. Anyway, I got in.