Ethan dives deep into the sea and enters the submarine Sevastopol. He easily opens the doors to swim into no-water chambers. There is no way he can do it because the pressure of water will hold the doors strongly.
Siloed missiles are shown with various national flags.
Neither France nor the UK have silo based nuclear missiles.
Pilot (and aircrew) parachutes do not have a reserve chute.
Hunt stages his dive on the Sevastopol from the USS Ohio where the commander has stated their current mission is to locate and identify Soviet submarines in the North Pacific. The USS Ohio is a missile boat whose mission would be to avoid all contact with other vessels. Become a ghost. The Ohio would not be used to launch a commando operation.
Whilst in Defcon 1 Lady President is calling US Strategic Command. In a matter of fact, in DEFCON 1 when nuclear war is imminent, US STRATCOM Commander would be one of many people surrounding POTUS giving her advice to use (or not to use) weapons of mass destruction.
The fire in the SOSUS station spreads much more rapidly than it would in reality. Some sort of accelerant had to have been used for filming purposes.
As the lights go back on around the world, it is simultaneously nighttime in both the eastern and western hemispheres.
The notion that transmitting antipodal coordinates would somehow trick the Russians into going to the wrong location makes no sense. While the Russians didn't know the exact coordinates where the Sevastopol sank, they certainly knew in which hemisphere it occurred.
In the final scene, Ethan exits the London Underground from an exit marked "Trafalgar Square" onto the famous landmark. There is no Underground exit at this location; the exit Tom Cruise walks out of is a public toilet. The sign has been added for the film as there is no Trafalgar Square Underground station in London. The closest station to Trafalgar Square is Charing Cross.
Kittridge says that the bombing of the Kremlin in Mission: Impossible - Ghost Protocol (2011) was specifically done to kill Ethan. This is not true; the bombing was done to cause the Russians to take a long time to notice that a nuclear device had been stolen.
At one point General Sidney orders a Marine Gunnery Sergeant on guard duty to surrender his side arm to him. No Marine would surrender his weapon to anyone, regardless of rank without proper procedure. The ghost of Chesty Puller could appear and tell a Marine to surrender his sidearm and the Marine still wouldn't do it. It violates their 5th general order.