On two occasions flights of English Electric Lightning fighter jets are seen under the command of a US battle group. Lightnings were never flown by the USAF, but only the British RAF and a couple of small nations. Of course, seeing this happens in the future, anything is possible.
Assuming that a submarine's control room plotting table would be bolted to the floor, the plotting table moves when Captain Crane falls against it.
Strings guiding the shuttle from the saucer can be seen.
The sound heard from the helicopter at 5 minutes in is the wrong sound for this type of craft. What is heard belongs to a Bell 47G-2 or similar as the sound of a tail rotor and 2 bladed main rotor is heard. The Boeing Vertol CH-46 has two 3-bladed main rotors.