Up until this point in the series, Artie has referred to himself as a Secret Service agent while in the field; after this point, he will refer to himself as an NSA agent.
When Artie answers the phone in H.G. Wells' hotel room, it's on speakerphone, even though all he did was pick up the handset, not pushing any buttons before or after.
When talking outside the music shop in Moscow, Pete tells Artie he could've mentioned on the plane why he had changed his name, as if it was new information to them, but Artie had already previously told them that it was to hide from the Soviets, not from their own government.
When Todd from the hardware store comes by the B & B to meet with Claudia and drop off the brackets needed for the warehouse shelves, they are in a large brown paper bag which he hands to Claudia and she sets them on the table between them. As they talk together and then also after he leaves and Claudia and Leena talk, the position of the bag changes as does the openness of it; it is sometimes closed and sometimes open as the scene changes back and forth in spite of the fact that nobody touches the bag after Claudia sets it on the table.
It makes no sense for the croquet balls to be a pair, croquet sets have 4 balls (one per player) and you don't usually hit one ball with another, you use a croquet mallet.
Charles II croquet ball is apparently "imbued with all the fury of a king", however Charles II was known for his extravagance, his affairs and his wigs, not his temper.
When Claudia shows Artie how to use his glasses to locate their case, she tells him he has to touch the temple tips together to close the circuit for them to work. When the others use them to find him later, the glasses are opened, meaning the circuit is open and they're not transmitting. However, it is possible that Artie intentionally touched the temple tips together before he dropped them thus setting off the transmitter and it was still working when Myka or Pete found them or that Myka or Pete did so to get the signal transmitting after they found them.
When Pete and Myka are suppose to be in Russia searching for Artie they agree to meet later and separate. As they do the camera pans to the left on the street there is a very quick glimpse of a vehicle's license plate. It is not a Russian license plate but an old style Ontario, Canada plate that is navy with white lettering.