Power Broker - S1-E3
Factual error: Every guard in the prison has the emblems of Hamburg on his/her shoulders of the uniforms. Zemo is said to be imprisoned in Berlin, so they are wearing the wrong emblems.
Character mistake: Bucky says he read The Hobbit in 1937 when it first came out. But in 1937 he was still living in New York, and The Hobbit was only available in the UK at that time, with fairly limited numbers printed. It wasn't available in America until 1938.
Power Broker - S1-E3
Continuity mistake: At the bar, the amount of liqueur in Falcon's snake drink increases between shots.
Other mistake: When walking through the Captain America display at the Smithsonian, Brody and Sam stand in front of a Captain America mannequin next to his shield. The mannequin is wearing boots of noticeably different heights.
Factual error: Walker has captain bars on his chest but is wearing lieutenant rank on his headgear in the photo. He is also missing the US Army tag on his uniform. Regardless of his actual rank at the time of the photo, the mix and match of rank and the missing Army tag would have been caught long before that picture was taken, especially with his friend in the photo being a sergeant major that catches uniform mistakes like that for a living. (00:34:58)
Chosen answer: The sleeve probably would have been ripped to shreds in battle anyway. He was just saving time by ripping it himself. Also, gunfighters would take off their coats in a duel or an arm wrestler rolling up his sleeve.