When Stan and Gennadi are eating pizza and watching hockey, they both have pizza in their hand and then after the goal is shown for one second, neither now has pizza. Also, there are two pieces left in the box before the goal, now there is only one.
Michigan scenes show front license plates on every car. Except personalized plates, the blue plates shown (used 1983-2007) were never placed on the front of cars.
Elizabeth tells Paige that the "Great Patriotic War" was World War 2. It isn't, and no Russian would ever confuse the two. World War 2 began in 1939 and ended in August 1945 and involved the United States, Japan, an many other countries. The Great Patriotic War began in June 1941 and ended in May 1945 and refers only to the Nazi invasion of the Soviet Union and its aftermath. It is inconceivable that a loyal member of the KGB would ever confuse the two.