Revealing mistake: When the team descends to the bottom of a hole and one female member is first wounded and then dies. After checking her pulse, the others pronounce her dead and begin moving out. As the last one is leaving, the "dead" actress lifts her head up off the ground and looks left. (01:13:10)
Visible crew/equipment: When Freddy falls over the banister onto the stairs near the end, notice the crash mat he lands on. (01:18:55)
Continuity mistake: When Peyton returns to his lab after its destruction, a burnt picture of him and Julie is on the floor. The destroyed picture shows half of Peyton's face missing. After finding a new place to stay, when the picture is put into the scanner, more of Peyton's face is visible. (00:28:10 - 00:30:40)
Plot hole: When the cop shows Pam the newspaper clippings they include what look to be fresh photos of the real Jason from the events of Part IV, yet no photographer is ever seen taking pictures of Jason alive in the previous films and most likely wouldn't live long enough to publish the photo. (01:25:00)
Suggested correction: The picture was taken from security footage at the hospital after killing the 2 employees.
It was never shown.
Visible crew/equipment: There is a shot of Holly crawling through a tight space with the camera facing her moving backwards as she crawls forwards (about 30 minutes into the film). In this shot, the wheel marks of the camera can be seen on the ground in front of her. (00:30:05)
Continuity mistake: After Sarah and Rios come out from the flooded cave, their clothes are bone dry even though there were almost fully submerged underwater. (01:01:25)
Factual error: How would the creature be able to re-enter the ship by climbing into the reactor piping? The reactor fuel system is not open to the living space of the International Space Station.
Continuity mistake: In the end, when Michael has grabbed Karen's ankle and is trying to pull her down into the basement, Allyson grabs a knife off the floor and stabs him. However, if you watch very closely, you can see she's already holding the knife in one shot before she grabs it. (It's the shot right before Laurie screams "I've got you!" to Karen.) You have to pay attention because it goes by quickly, but you can see Allyson lifting the knife into frame for a split second, even though she doesn't pick it up off the floor until about 15 seconds later.
Character mistake: Selena Gomez drives into town in a fully restored 1967 Pontiac Le Mans, which is a long box shaped car. However, when confronted with Bill Murray and Adam Driver, Adam Driver complements her on her 1968 Pontiac Le Mans, which is a completely redesigned shorter curved body style. The two look nothing alike.
Continuity mistake: When Gwen and the police are at the house containing the buried bodies of the victims while Finn and the Grabber are fighting, the Grabber's van is not in the carport. However, when Finn exits the house, and he and Gwen embrace, it is. (01:28:03 - 01:34:27)
Plot hole: Peter goes into Carl's mind to save Catherine. When he's in there he sees the tank with the water nymph. On the tank are strange symbols which provide the FBI with the clue needed to find the latest victim. Makes sense so far. But, go back to the scene just after the FBI have captured the comatose killer and are looking in his basement. The FBI are looking at the contraption that the killer uses to suspend himself over the victim. On the contraption is the same symbol seen later on the water nymph's tank. Why didn't the FBI follow up the symbol then?
Revealing mistake: You can see wires in the scene where Regan levitates above her bed in the exorcism scene, lifting her up. The way they pull on her dress is also visible.
Revealing mistake: When Henry's mom is pushed off the cliff her cable is visible.
Continuity mistake: During the paper cut scene, when Bella is thrown against the table, there are 5 vases of flowers and 3 sets of candles. After she hits the table she lands against one of the vases and knocks the 2nd one to the floor, but the next shot shows only one vase left upright. (00:14:50)
Continuity mistake: When Hauser picks up the Admiral, after the Brain Bug attacks them, she has no yellow alien blood on her as she's being pulled up, but when the shot changes, as she's standing up, she now has blood on her face and uniform.
Factual error: The moon is full every night the highschoolers are at the lake house.
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?
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.
Continuity mistake: When the white boy is standing outside the car talking to the gangbanger inside the car, the gangster's sunglasses disappear depending on the camera angle.
Factual error: The aircraft used to fly to Shangri-La is a Bristol Beaufighter, a single seat long range fighter bomber. It was not a transport aircraft and could not carry passengers. Some variants of the Beaufighter were two seaters but none at all could be used as small airliners as they do here. It is not possible to simply reconfigure the aircraft as shown here - the centre of gravity, balance and trim would all be thrown out. The resulting jury-built aircraft would be uncontrollable in flight.
Continuity mistake: When Christine is fixing herself after the guys smashed her up, her front end is shown to be perfectly straight for its entire width and the hood is missing. Near the end of the scene, the front end is shown crushed and unfolds again, and the hood appears from nowhere.