Continuity mistake: Lane is listening to the virologist on their flight to South Korea. Sunlight alternately appears and disappears from his torso. (00:37:00)
Audio problem: While Gerry is talking to the C.I.A. Agent, a side view of the agent's face is seen as he says "That's the man in Jerusalem you want to see." The movements of his mouth and cheek does not appear to match up to what he is saying at all. (00:43:55)
Character mistake: When Gerry is in Camp Humphreys, the leader of the group there tells him to make sure Maverick keeps the nose of the plane pointed towards the sun. However, since they're going to Jerusalem, and it's night time, they would likely be headed West, therefore not towards the sun but rather in the opposite direction. (00:48:40)
Continuity mistake: Lane is in Jerusalem talking with the Mossad officer just as the zombies are about to cross the wall. In all shots from the front, the left side of the Israeli's face is bathed in sunlight while the right side isn't. In shots from the back, the entire right side is now bathed in sunlight. (01:00:00)
Continuity mistake: The plane that Brad Pitt gets on board in Israel is a Airbus A310 from the exterior shot, but the interior scenes take place in some kind of Boeing widebody aircraft, as noted by the Boeing signature interior, and more significantly the design of the overhead compartments. (01:07:55 - 01:16:50)
Other mistake: When Gerry and Segen are in the plane towards Cardiff, a zombie attacks a flight attendant and a whole section of the plane. Despite the fact that what's happening would be obvious to the passengers of that section, no one even tries to escape the attack by going to the back of the plane, in Gerry's section. Gerry and the passenger have the time to pile up luggage without anybody from the front section interfering. (01:16:40)
Continuity mistake: During the outbreak on the plane, the same guy is seen running from his seat twice after each of the baggage walls is broken down. (01:18:00)
Continuity mistake: In the scene after the plane crash, Brad Pitt is hanging by the waist with his face towards the ground and blood dripping from it. When he looks up, there are blood trails running down his face rather than towards the front of it. The blood trail should be going from the cuts toward the front of his face, not towards his chin. (01:20:45)
Factual error: The first time Brad Pitt leaves the aircraft carrier he is on an Antonov An-12. I haven't been able to find any evidence of an Antonov An-12 being able to take off or land on an aircraft carrier. The minimum runway distance for takeoff is 1500 m, and A Nimitz class carrier is only 325 m. (01:22:00)
Suggested correction: Most planes, if not all, that don't have vertical takeoff capabilities can't land or take off on aircraft carriers. That's why systems are in place to launch planes up to speed for take off and to catch or stop planes upon landing. Even jets that are 5 times faster than a An-12 need help with take off and landing.
A C-130 did multiple take-offs and landings aboard a carrier WITHOUT any assistance, so yes, an AN-12 might have been able to do the same thing: https://theaviationgeekclub.com/the-story-of-how-the-c-130-hercules-became-the-biggest-aircraft-to-land-on-an-aircraft-carrier/.
Continuity mistake: The girls are kicked off the USS Argus when the military thinks Gerry is dead. Thierry sends them off on a helicopter. As they board the helicopter, we see a bridge structure behind Thierry, who is looking down the length of the ship at them - and also behind them is 'the same' bridge structure. It has been shown, in several shots through the film, that the Argus has just one such structure. (01:25:20)
Continuity mistake: As Gerry, Segen, and the head of W.H.O. Are walking through the cafeteria of B-Wing, the head of W.H.O. accidentally kicks a can of Mountain Dew which slides across the floor. Just after this, the other W.H.O. Scientists are shown looking at the three in 3 different security monitors. The movements of the three in each monitor is different and does not match up if you look closely. (01:29:30)
Continuity mistake: Segen and Lane are leaving the WHO facility. Just before walking through the gate, Segen's purple hood is up. In the immediate next shot, as she steps outside, her head is bare. (01:52:25)
Continuity mistake: When Gerry reunites with his family in the Nova Scotia Safe Zone, the boy gets into the family hug, with his sweatshirt hood down. The camera angle changes and it is up. (01:55:35)
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: After the family reaches the apartment and everyone is listening to the radio, the light for the fish tank is on while the power is out.
Factual error: Gerry and Thierry use Iridium 9555 satellite telephones to communicate with each other. These telephones require a clear view of the satellites in order to operate, however both are able to communicate even when a clear view of the sky is impossible, such as inside an aircraft or in the bowels of a warship.
Continuity mistake: After escaping from South Korea, Gerry calls his wife who is in bed with one of her daughters, Connie. Between shots as Karin answers the phone, Connie's arms change position.
Factual error: Atarot Airport in Jerusalem has been closed since 2000. Even when open, it was a tiny regional airport only designed only for eight propeller passenger planes, not the major international airport depicted.
Continuity mistake: In the scene where the plane decompresses, the pilots are shown putting on air masks. Seconds later just before the plane crashes, the pilot is again shown, this time without an air mask.
Factual error: When flying from Korea to Israel, a nuclear explosion happens in the distance, the plane is hit by a shockwave, and then the phones go dead. The EMP from the nuke would have taken out all electronics first, not last. And they certainly wouldn't work again, especially a sat phone designed to receive radio waves. It would have probably also taken down the plane.
Suggested correction: There's no evidence this was a nuclear explosion. The mushroom cloud effect is most commonly associated with a nuclear explosion, but any sufficiently energetic detonation or deflagration will produce the same effect.
Except in this case, the explosion shown is a copy of a real nuclear detonation (Operation Crossroads, underwater), so it's obviously meant to be a nuclear explosion.