Brody prays when sun is rising. In Islam, morning prayer must be finished before sunrise, and next prayer is at high noon.
Marine Security Guards (MSGs) assigned to the Embassy in Pakistan do not perform bag inspections of visitors. This is done by Local Guard Forces (LGFs). Nor would MSGs be marching around the compound.
Brody and other US Marines are referred to multiple times as "soldiers", including by a character who claims he was with a UDT (underwater demolition team, a "sister" team to the Navy SEALs who have to pass the same training). It's a point of pride among US Marines that they are marines, not soldiers. Brody refers to the UDT character as a soldier. As Naval personnel, he would be a sailor.
Brody's uniform ranking changes several times throughout season 1. At highest rank on service uniform he has the ranking of Gunnery Sergeant. In dress blues to attend dinner, he is shown with only a Sergeant ranking. In reference to repeated naming of ranking, he is referred to as Sergeant, if the Gunnery Sergeant rank is true, everyone would refer to him as Gunnery Sergeant or Gunny for short
Throughout the first season, Carrie continuously calls the hijab headscarf a burka. As her character has been on service in the Middle East before, it is extremely unlikely she would confuse these terms.
The close personal relationship between Brody and Mike blatantly violates military regulations against fraternization between enlisted men and officers. A case this flagrant would almost certainly lead to a court-martial.