There is a real Weaselby; Cole took his place.In the cutscene when Hatsworth and Weaselby first meet, they clearly recognize each other; Henry mentions that Weaselby's dues to the Pompous Adventurer's Club are overdue. Clearly Weaselby planned to get the treasure to pay off his debt, until Cole made a robotic fake to take his place.
Lady D is Death.None of her husbands left her by choice, and she's hanging around a graveyard...
The Master Piece just grants its wearer multiple powers, and presumably access to the Puzzle Realm as well.Each piece of the Suit grants a different power; the Hat reduces the wearer's age, for example. With the possible exception of the Shoes, any one piece could open up the Puzzle Realm, including the Master Piece. That's presumably how Weaselby vanishes into his hat; he's hiding in his own "Touch Screen". Also, Cole didn't take into account the fact that he only had just the one, whereas Hatsworth had all the others.
The age-reversing effect of the hat was a failsafe made by its creatorOlder, mature wearers have their vigor restored, but younger people who would be more likely to use the hat's power for immature ends are rendered infantile and powerless.
The Captain is THAT CaptainHe traveled to this game's world to get away from the madness of Soleanna, but the memories stayed, and over the years they slowly drove him insane.