Opening scene shows a Catholic Church with the manger outside and snowing. It is obviously taking place during the season of Advent/Christmas season. The church is dark and we see the candles and a statue of the virgin Mary. However, there is no sign of an Advent wreath near the altar or hanging up, which is the focal point of a Catholic church during this season.
During the nighttime shootout on the river bank, the usual Browning M1917 water-cooled machine gun manned by Robert Lackey suddenly alternates with the M2 .50 cal machine gun. There was one shot when the muzzle flash lit up the scene that you see the long barrel of M2 .50 cal, not the M1917 with the cylindrical
water-cooled jacket. (This is another machine gun team, not Leckie and Chuckler)
The map of the world showing the Axis powers' extent shows Iceland as Nazi occupied - although a plan to invade Iceland was drawn up during WWII, the island was never invaded or occupied by Germany. The British occupied the officially neutral Iceland in May 1940, replaced by the USA in July 1941. So at the time of Episode 1, December 1941, Iceland was occupied by the USA, not the Nazis.
At around 8:15 Col. Chesty Puller makes a short speech to a number of US Marines. Behind him is a huge world map showing Nazi and Japanese conquered territories. Clearly it shows Iceland as being one of those conquered areas and that simply was not true. Iceland was never held by the Germans in WWII and they didn't even try.
The depiction of the Eastern Front on the map in the war room mixes up situations from 1941 and 1942. In the north, the front line is correct: the Tichwin bulge southeast of Leningrad was about to collapse in the second December week, but that probably was not known in an American war room about the same time. Also, the Soviet winter offensive has just begun, so the front line in the center is still being showed correctly near Moscow. But in the south, the German advance shown is definitely that of late summer 1942, with the Wehrmacht approaching Stalingrad and the Caucasus. In winter 1941, the front hadn't yet extended beyond the sea of Azov, which on the map is the tiny "lake" east of the Crimea.
The morning after the battle with the Japanese, a group of stragglers emerge from the jungle and engage the Marines. In one close-up (around 43.21), the rounds in the Marines' machine-gun belt can be seen to have crimped ends which means they are blanks.
Right at the start of this series when the marines first land on Guadalcanal. They are surprised to see no enemy resistance, and as they start heading into the jungle right at 29:39 they come across 3 mutilated Marines. How is this possible when the platoon of marines were the first ones to land on Guadalcanal?
At approximately 25:45, a Marine is shown carrying his M1903 rifle at 'low-ready' with the stock near his left shoulder and the rifle pointed to his right. This would be unusual, but not unheard of. However, the bolt handle on the rifle is shown on the left side of the weapon indicating a flipped shot.
The large map shown on the wall is a wide-format print but this technology was not available in 1941, when this is set. The only technology available to produce this extremely large sized map would have been using a collage of lithographs (but it has no seams) or hand-illustrating (but it's clearly not hand drawn and uses typefaces for the markings).
In episode 1 and in photo 38 of 156 above, showing William Sadler as Chesty Puller, the map of Europe on the wall shows Iceland as being occupied by the Germans. Iceland was occupied by British forces to prevent any German occupation. US forces took over the occupation as neutral forces, prior to the US involvement in the war, and remained there until 1944.
At 6' Brasília appears like Brazilian Capital, but this city only was constructed after 1955. At the WWII time Rio de Janeiro was capital of Brazil.
During Lt. Col. Puller's presentation in which he describes the areas conquered by the Japanese, he mentions Hong Kong but he clearly points to Hainan Island, almost 400 miles to the southwest of Hong Kong and about 40x larger.