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?
Continuity mistake: Just before the garbage truck mows down the policeman and continues through traffic, we see behind the cop a street full of stopped cars, and a building less than 100 yards away (the street starts at a T-intersection). There is no truck, and no way for it to have gotten onto the street at the speed it's going. (00:07:05)
Continuity mistake: Just before the motorcycle cop smacks their car's mirror off, Gerry says to Karin "You're not British anymore baby." As he does, she raises her coffee cup to her face in the frontal shot. It then cuts to an angle to show him, and suddenly her arm is down, where she raises her coffee cup to her face again. (00:06:05)
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.
Continuity mistake: The family car's side view mirror gets knocked off by the police motorcycle and falls in the space between Brad Pitt's car and the one in front of it. When Pitt bends to pick it up, it has moved further up the road, next to the back tyre of the car in front. (00:06:35)
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: 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)
Revealing mistake: When the family's car is about to be hit by the ambulance as they try to escape downtown, Gerry is driving and his wife is up, leaning over the passenger seat talking to their daughter. The camera angle changes to a view from behind, and as the ambulance is about to hit, we can see there is nobody in the car and the ambulance driver is wearing a safety helmet. This is viewable in just a few frames. (00:08:00)
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: 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: While playing games in the car, at one point a white van can be seen out Karin's window. The shot cuts to the daughters in the back as Gerry asks if it "is kept in a pen all its life and harvested for meat?" then back to the wife, and suddenly the van has vanished and you can see a couple people out the widow. The shot change was not long enough for either vehicle to move enough. (00:05:30)
Other mistake: When the family's car is being hit by the ambulance, we see Gerry's head bouncing around by the impact in a close-up. Camera angle is from inside (approximately from the centre armrest). We see out his window, and there is nothing there. The camera angle changes to an exterior shot, and its clear that the position of the ambulance hitting the car should have made the ambulance visible from the interior camera. (00:08:05)
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)
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.
Continuity mistake: After the motorcycle cop knocks off his car's mirror, Gerry gets out to pick it up but doesn't shut his door fully, leaving it about half open. When the guy in the red car gets out and asks if he is all right, the door is suddenly not open as wide. Then it is back to how wide it was in the next shot. (00:06:30)
Continuity mistake: When Gerry and his family are escaping the first wave of the zombies they encounter, they steal an RV and make a run for it. While trying to shake a zombie off the vehicle, Gerry smacks into a taxi, bending the bumper and busting out the headlights on the RV's left side. A little bit later in the film as they are traveling down a highway, they pull to the side of the road to try and find Rachel's spare inhaler, and the front of the RV is shown. The bumper is still bent, but the headlights are now fixed again. (00:10:25)
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.
Other mistake: As Gerry enters the aircraft carrier command centre, we see screens with "world projections" of remaining uninfected populations by continent. They are counting down (illustrating the quick spread of the disease). However, the countdown speed of all of them, is approximately 3% each second - which means humankind would have no more than 20 seconds at this rate. The countdown speed should realistically be about 100,000 times slower than that (about 3% each day), which would still be the fastest pandemic spread the world has ever seen. (00:30:15)
Suggested correction: The countdowns are only seen for a couple of seconds, so there's no evidence that the countdowns are constant at the same rate. They may be updated in bursts as data is received. They may just show a rapid initial reduction that slows then flattens out. They may be wrong (because how could they be correct). Lots of explanations why this might not be what it seems.
Continuity mistake: There are two different transport planes shown when flying to Korea and again to Israel. The first shot taking off from the aircraft carrier is an Antonov AN-12, but then other shots show a C130 Hercules. Some of the inside cockpit and cargo area shots are sometimes a C130 cockpit (can tell by the types of windows and cargo seats) but others times a different plane. (00:32:35 - 00:49:20)
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)
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.