Continuity mistake: In the opening scene, the two girls open the door to their parents' room. Karin's pillow is in the shadow. Sunlight suddenly appears on it just as the two girls are about to jump onto the bed a few seconds later. (00:03:25)
Continuity mistake: When the movie starts with the family lying on the bed, Brad Pitt's arms keep changing positions between frames. (00:03:30)
Continuity mistake: As Gerry is handing his daughters a plate of pancakes, he says "You mean a stuffed animal puppy? 'Cause that we can do." As he says this, the camera angle changes and suddenly one of the girl's hands can be seen up in the air together almost in a praying position. The camera cuts again and her hands are back down. (00:03:55)
Continuity mistake: Gerry is standing by his stove where he is cooking pancakes, looking to his right at the TV. Above the TV is a cabinet with a purple sticker on it. As his daughters and wife get up to leave the room, he is suddenly standing next to this cabinet. Then, only a split second later, as he says "Hey! Plates in the sink!" he is suddenly back in front of his stove, which is a few feet away. He moves to these spots instantaneously. (00:04:55)
Continuity mistake: In the car, before the first zombie attack seen in the film, the family's two daughters are in the back seat as they are playing a family game. The youngest one, Connie, leans over to whisper in Rachel's ear, and it is a profile shot of her face and arm, her neck and some of her chest visible. It cuts to a different angle as she is sitting up, and suddenly she has what appears to be a large stuffed bunny or mouse in her arms, with its ears covering a lot of her face that was not visible in the previous shot. (00:05:10)
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)
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 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: 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)
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: The first guy that Gerry watches turn into a zombie, and figures out the transformation takes about 12 seconds, he sees go to his car first. The guy opens his door and his right arm is down towards the center of the door. The angle changes and suddenly his right arm is up towards the top of the door. (00:09:05)
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)
Continuity mistake: After the Mexican family lets Gerry and his family into their apartment, they sit down on the couch. As Gerry leans over and kisses Rachel's head, Karin's hand is over Connie's left shoulder. The angle changes and suddenly Karin's hand vanishes. (00:17:30)
Continuity mistake: As Gerry enters the ship's command centre the first time, when he starts walking across the floor, we see a bearded man in the background, at the stairs against the wall, talking to someone. The camera angle changes, and he is now sorting through papers perhaps 60-70 feet from where he was just standing. (00:30:30)
Continuity mistake: When Brad Pitt and his wife say goodbye, her right arm is extended, but a frame later it's bent and holding Pitt's hand. (00:32:25)
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)
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)
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)
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)
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.