During the discussion in Dooley's office, when Carter steps out of the room to set up the meet with the Howling Commandos, Dooley says he's getting calls from the Vice President of the United States. But in 1946, there was no Vice President. Harry Truman succeeded to the Presidency upon Franklin Roosevelt's death in April 1945, and there was no Vice President until Truman won his own term with Alben Barkley, who took office in January 1949.
At the beginning of the mission they find a film projector. Agent Carter stops the projector, with the light still on, to read some subliminal messages embedded in the film frames. Doing this would melt/burn the film in a few seconds, but it does not.
A one-time pad can't be "cracked" (unless the users ignore the "one time" bit and keep using the same pad for multiple messages) by any form of crypt-analysis, you can only decode it if you HAVE the relevant pad. Even then, unless you have memorised every letter on the pad, you can't decode it "in your head" nor could you recognise, say, "map co-ordinates" in the cypher text.
Capt. Thompson jumps from the plane before the light changes from red to green.
Before parachuting into Poland, the team does not connect the static line of their parachutes to the plane.
While working on the code, the team keep on mentioning Belarus. However, Belarus wasn't known as Belarus outside of the country until it gained Independence from the USSR in 1991. The country, at the time, was most commonly known as White Russia, which is a translation of its official name at the time, Byelorussia.
When Agent Carter is decoding the message, she reads out coordinates of "53 degrees, 72 minutes North and 27 degrees, 37 minutes West." There are two errors here:
1.) In angular measurement there are 60 minutes in a degree, just as there are 60 minutes in an hour. To give a longitudinal coordinate including 72 minutes is incorrect, as it would be to give the time as 72 minutes past the hour.
2.) The coordinates given are not in Russia, but in the North Atlantic ocean.
However, If one assumes they really meant 27 degrees, 37 minutes *East* , then that would put them somewhere between Vinius and Minsk, which is where they were supposed to be.
When Carter and the team are about to make the parachute jump, the static lines are still wrapped around their chutes. They should have been hooked to a cable in the aircraft to allow the lines to deploy and eventually open the chutes.