When West goes into the abandon store front the broken glass panel in the door is a different shape from the inside.
When West goes into the abandoned storefront and uses the key to enter the 'barrel' into the room where Harry Holmes is carving a printing plate, West inserts the key into the slot, but never really turns it, even though he acts like he does.
While searching Bronzini's trailer, West opens a trunk and pulls out a box with a note and key. During this time, he casts two shadows - one on the wall and trunk in front of him, and one on the wall to his left (we see it as the back wall). This would be impossible given that there was only one light source - the lantern he lighted when he entered.
The U.S. Mint (which is a bureau within the Department of State) does not print paper money, it mints only coinage (i.e. specie, "hard money"). The Bureau of Printing and Engraving controls the production of paper bills of various denominations and is within the Department of Treasury where West and Gordon are employed as agents.
When West is in the storage room trapped under the large crate, he uses his double-barreled Derringer to first fire the barb device over the beam, then uses it to fire the barb into the crate itself, thus creating a crude pulley to lift the crate. But when he fires the barb device both times, he uses the top barrel of the Derringer - without reloading.
Near the end as West and Gordon are following Lennox, West tells Gordon to cover the back. Gordon nods and walks off (not speaking). As he walks off, it's briefly evident that the actor is not Ross Martin.
While at the circus, Jim buys Artie a sausage sandwich and refers to it as a 'hot dog', a term not coined till around 1900.
When Arty is in the emporium the lighting quite visibly shifts as Arty and Mrs. Moore move about.
At the end of the second act (26:20), Artemus falls behind a counter while several men shoot at him. At the beginning of the third act, Artemus is all the way across the room.
The pistol that Lennox has, the cylinder swings out. That feature was not invented until much later. At that time you loaded one round at a time through a loading gate, or dropped the cylinder out of the pistol.
When Jim and Artie are talking to Lennox in his office about a possible theft of ink and paper, the shadow of a boom mic is visible on the wall behind them.
About halfway through the episode, West is accused of libel for making the claim that counterfeiters are being supplied with paper. Libel is written defamation; the accusation should be slander, which is spoken. But, in either case, slander and libel cannot be made to the one who is accused - it must defame them, which it cannot, if made to the accused.