Factual error: An An-12, the aircraft the main character uses to fly all over the world, has a maximum range of about 3,500 miles. Hardly enough to fly from the US to South Korea or from South Korea to Israel. The An-12 also miraculously transforms into a C-130 in a couple of filler scenes. And why is this ex-Soviet aircraft marked in USAF markings, assigned to McGuire AFB?
Factual error: The old percussion double barrel shotgun is firing modern shotgun shells, which would never work or fit in that gun, and furthermore are also made of plastic, wrong for the era.
Other mistake: During the final shootout in the saloon, young Joey yells, "Shane, look out!" Alan Ladd whirls around and his gun goes off. But the gun isn't pointed anywhere near the bad guy who is standing on the second floor balcony. Shane more than likely shot the furnace that was off to the right. Yet, the bad guy still manages to do a face plant on the barroom floor.
Factual error: It is not possible that Pitt could have gone up to the ship when it was already blasting off. There was literally fire in the tunnel.
Suggested correction: It was a bit confusing, but what I saw was a shower of sparks or hot particles and some fumes, and no fire in the tunnel until he was through the hatch. The makers may have been influenced by seeing vapour prior to a rocket launch, and then some rockets use a shower of electric sparks to ignite the engines. It was implausible, but no fire in the tunnel.
Revealing mistake: When Claude and Ray are at the retirement home getting ready for bed, as Ray lifts up his shirt you will see where it looks like he's wearing a fat suit underneath. He can also be seen pulling it down to his waist.
Visible crew/equipment: Near the end, when Rodmilla and her daughters are requested to appear before King Francis, in the first shot facing the king and queen, there is a beige narrow mat in front of the large patterned rug, which is not in previous or following shots, specifically used for crew/camera tracking (fullscreen version). (01:50:30)
Continuity mistake: When Fox and Wesley first jump off the building onto the moving train, Fox has on boots with heels, but when she lands on the top of the train, her boots are flats.
Factual error: Denham is obviously shooting a sound film - he has a sound recordist with him along with the bulky and awkward recording equipment typical for the era, and they discuss the problems of recording dialogue on board. But not once do we see him filming with sound. We see the crew recording dialogue - synchronised sound, recorded on location, which is utterly impossible given the equipment they have and the circumstances under which the fim is being shot. We never see a microphone, a boom pole or a tape recorder. His camera isn't even 'blimped' - soundproofed - and it's handcranked, which makes a racket. They can't be planning on adding the sound later - why have the sound recordist and his bulky and heavy equipment there with them if they are? We see the crew recording dialogue - synchronised sound, recorded on location, which is utterly impossible given the equipment they have and the circumstances under which the fim is being shot. The whole point of post dubbing dialogue is that you don't need a sound recordist in the first place.
Other mistake: One visual that has always bothered me that I could not find in your list were the scenes when the mother ship first appears. It's enormous scale appears to dwarf Devil's Tower and the whole surrounding area actually, but when it moves over to the "landing strip" area and begins to rotate 180° (right-side up?), it suddenly seems to shrink to a much smaller size and mass during the slow revolution. On its originally-seen scale above/behind the tower, one would think that either the great ship's outer prongs would have been torn off, or more likely the impromptu landing site and most of Devil's Tower would have been destroyed as the huge craft rotated itself. The visual scales just do not stay consistent throughout the film's climactic final act.
Factual error: At the point where Jared Leto is released from prison he is picked up in a 1960 Black Chrysler Imperial. As the camera rises it shows the date as 1954. The car is used a few more times in the movie.
Factual error: The movie takes place during the 1990s, but throughout the movie cars are shown with Illinois license plates with "Illinois" written in cursive. These plates were not introduced until 2001. (00:24:45)
Factual error: When the B-52 is flying low over Russia, the shadow on the ground is a Boeing B-17G, a World War II propeller driven bomber.
Continuity mistake: When the police are moving in to arrest Dzhokhar Tsarnaev, the shot shows an FBI SWAT officer grabbing him. But in the next cut, the officer changes to a Boston Police Officer while changing back to a FBI Officer in the following shot.
Revealing mistake: When the new Pope is announced, the New York Times headline is shown, and 'Copyright 1990 The New York Times' is visible. The event took place in 1979.
Continuity mistake: The morning after Vivian's first night at the hotel, room service brings breakfast. When she's eating a croissant it cuts to a shot of Edward, and when it cuts back to Vivian she's eating a pancake which goes from being half done, to being a whole pancake again. (00:30:55)
Other mistake: The first time the cops come to the house, Connie and her son see them from the upstairs window, and start walking downstairs. It then cuts to the cops in the living room looking around, and Connie walks in and introduces herself. Her husband is not home, so did the cops just walk into in the house without knocking?
Suggested correction: I'm pretty sure the housekeeper let them in if you listen carefully.
Continuity mistake: When Eliot Ness is at Jim Malone's house for the first time, Malone's collar is opened, then buttoned, then open again.
Revealing mistake: When Sonny is punching Carlo under the spraying fire hydrant, he misses an audible punch by at least six inches. (01:43:30)
Continuity mistake: Towards the end, Denzel drives his car into 2 cars trying to get Ethan off the hood. After that scene ends you see Denzel driving the same car and the front and rear of the car look nothing like they did when he was crashing into the cars.
Continuity mistake: During the scenes at her Mother's house, Kate's hair goes from blonde to REALLY blonde.
Suggested correction: Can't speak to the second half of your paragraph (should really post as 3 separate mistakes) but as for the first, a range of 3500 miles, aircraft such as the kc-135 exist and aerial refueling is fairly common place. Considering it's a mission supported by the acting UN Secretary General to stop a world crisis, resources could have been diverted for refueling.
The initial launch from the carrier is a C-130 which can do this (if empty, minimal fuel, has the full length of the flight deck and the carrier is steaming full ahead into the wind). It then morphs into an AN-12 and back to a Hercules. They make the point that this small fleet is what is known to remain of allied forces so not sure where any tanker support will come from. Many movies have ridiculous range issues with aircraft anyway.