The alien in the pod is eventually released to be Kayla, which is hard to believe since Kayla wasn't trapped in a pod when she was jettisoned from the alien ship. She was freely thrown from the ship when Sara released the rest of the pods, to prevent Sara from being pulled out of the ship by Kayla. Even the rest of the Legends saw the pods disappear into the Temporal Zone stream, Kayla was shown to disappear with just herself.
The nuclear football is presented as a control board which enables a single person to simply press a few buttons and thereby order missile launches. In reality, the football just contains a series of codes, which must be called in/transmitted by at least two separate individuals in order to authorize a launch.
General Kilgore is shown wearing Infantry branch insignia on his blouse lapels. General officers do not wear branch insignia. The only insignia on their blouse lapels is "US".
Both the Air Assault Badge and the Pathfinder badge were not authorized for wear at this time. The metal Pathfinder badge was not made until 1968, and the Air Assault badge was not officially adopted until 1978. These events occurred in 1962, six years before the metal Pathfinder badge was made and 16 years before the Army officially adopted the Air Assault badge.