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?
Revealing mistake: When Nica crawls to the elevator to escape Chucky, she bends her legs in order to shut the elevator door. She is meant to be a paraplegic and have no feeling in her legs.
Continuity mistake: When Fin stops near the school bus, the bus number changes from "278146" to "278245" during the rescue of the kids.
Revealing mistake: When Tommy Ross gets killed and Carrie looks at his supposedly lifeless body if you look closely you will notice he blinks revealing that he is actually alive.
Continuity mistake: They trick one witch with a record recording of a boy that is lost. H&G are thrown down by it and the witch picks up a branch and begins zooming off like it is a broom. Hansel is behind her, yet despite the witch traveling faster than he is running, somehow he suddenly gets in front of her and smacks her with his gun. (00:31:10)
Audio problem: In the scene where the polite leader is addressing the Sandins for the first time he takes his mask off and starts talking to the camera but he is muted until James unmutes him. If you watch the polite leader while he is muted he mouths the words "Mr. And Mrs. Sandin" before he is unmuted. However when he is unmuted we hear him finish that statement by saying "Mrs. Sandin" again. (00:33:25)
Factual error: Near the end, Mia discovers the chainsaw in the toolshed. Glancing around, she sees a clear plastic bottle of sparkling bright blue fluid that looks like window cleaner but is marked "Chainsaw Gas." Real chainsaw fuel is a 1:40 mixture of 2-cycle motor oil (greenish-black in color) and gasoline (pinkish-amber in color), which produces a fuel that is olive-green in broad daylight. In the darkened shed where Mia finds the chainsaw, the "Chainsaw Gas" in the bottle should not appear sparkling blue, but should appear very dark green, even with artificial light behind it.
Continuity mistake: When looking at the dog from the bedroom window, there are 2 windows in the room facing the dog. When we see the window from outside only 1 window is lit up, despite both being in the same room. (00:11:30)
Continuity mistake: John is taken down by a trap set by Hansel and puts a big hole in his back. He gets back up and is shot down by the Sheriff. As he is getting up a second time, it shows his back. There is no hole in his back or shirt before the Sheriff finishes him off. It also shows no hole or blood on his back in one shot when he is chasing with the bear trap before he is shot the first time. (01:16:45)
Other mistake: Everytime it shows a window it is morning, even if the scene is supposed to be at night.
Continuity mistake: We discover that because time is non-linear in the Further, it was Josh's spirit who was the one knocking on the door in the original film. However, the knocks we see Josh make in this film don't quite match up with the original film. In this film, he knocks many times frantically, whereas in the first film, we only hear a few faint knocks in rounds of three. (The film tries to mask this by inter-cutting footage from the first film to illustrate the point, but it still doesn't quite match up).
Continuity mistake: In the scene where Heather runs through the fairground she grabs onto the back of the Ferris wheel seat and is taken up in the air. Leatherface throws the chainsaw at the cop and the next shot shows Heather coming back down the other side of the Ferris wheel. But she should be on the inside of the carriages as it just spins round on an axis. These kinds of wheels don't "twist" at the top otherwise the rider would just be staring at the inside of the wheel the whole time.
Continuity mistake: When the group of girls first pulls up to the gas station and talk with Barney, the shadow cast by the sign is much longer than moments later when they leave. (00:11:15 - 00:12:50)
Other mistake: When Odd Thomas and Stormy are riding on his moped through town to stop the massacre they pass a car dealership. All of the letters on the dealership signs are reversed except for a large lighted awning sign that says "Welcome to quality." The normal and reversed signs appear at the same time in the same frame. (00:48:00)
Factual error: They inject the infected Brunel with antibiotics to fight the bacteria, however there is no pulse and that means the antibiotic can never spread through the body, as there is no blood circulation.
Visible crew/equipment: In the opening close up shot of the eye, you can see a reflector screen reflected in it. (00:04:10)
Continuity mistake: After her dog chases away his goons, Agnes has Carlos hanging upside down with a water hose. The first part, he still has his shirt tightly tucked in his pants. But when she says "Fucking is just not on the menu" his shirt is suddenly out and around his head, his chest exposed. But then it is tucked back in again moments later. (00:44:45)
Continuity mistake: Before Iggy gets his horns, he got a shot of whisky in the bar from the blonde girl. The level of whisky changes between shots.
Revealing mistake: When the doctor intercoms the colonel because he cannot get into the medical room for the pregnant woman having nightmares, the camera shows the bridge intercom, which is simply the back end of a desktop computer with a couple of knobs stuck on it.
Factual error: Tilda Swinton requests a flight from Detroit to Tangier, Morocco, but it must be an all-night flight since she and Tom Hiddleston are vampires. They end up accepting a flight connecting through Madrid. Flying west to east, it would be impossible. If they leave their home at sunset, say 6 p.m. Detroit time, and take off around 8 p.m. Detroit time, it is already 1 a.m. In Spain. Toss in the travel time and they will be landing in Madrid in the early morning, which would kill them.
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.