During the shootout at the abandoned shack, the scenes alternate between daylight for the exterior shots, and nighttime through the windows during the interior shots. Update: It is also possible the camera operator forgot to install the appropriate filter (or adjust the lens) during the shooting of the outside scenes, and by the time they processed the film it was too late to re-shoot them.
When Artemus (disguised as the Count) is fixing dinner, he mentions a couple of times that he is preparing chevapchichi (a traditional dish in the Balkan countries) at the table. A bit later, when Artie is trying to put one over on Stark and Kelso, Countess Zorana tells him to come back to the table because "...the chevapchichi is boiling." Chevapchichi is made from two kinds of seasoned beef and grilled, not boiled.
The program indicates that Bosnia was an independent country during this time. But during the Grant Administration, Bosnia was still part of the Ottoman Empire, and would not, in fact, become an independent country until 1992.
After Artie, as the Count, shoots the Sheriff and the deputies, he cracks a walnut with his "silver" hand, but the underside is not solid, and there is an opening at the heel of his hand.
At 32:06 when Draja is screwing his hand back into his arm, you can not only see the black thread holding the fake arm in place, but also the safety pin holding the thread. Really bad editing back then to not catch this kind of obvious visual mistake and not doing a retake.
As the train carrying Artemus Gordon travels across the countryside, a modern day cross-country electric power line is clearly visible in the background.
When asked what crime he committed the count responds, "They caught me cheating at tiddlywinks." The patent and trademark for this game were not filed until the late 1880s, a few years after the time of this episode.
No one seemed to notice during the opening sequence that West, who so prominently took possession of the Count, disappeared behind the box, and never got on the train.