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: 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: 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: 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)
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: 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: Three times through the movie, we see the command centre on the ship with people running around. Based on several people's movements, it seems that all these scenes were taken at one time, from different angles - too many of the same people moving the same way, carrying the same things.
Continuity mistake: When Brad Pitt is changing the dressing for the female Israeli soldier on the Belarus plane, the man in the seat next to her appears, disappears and re-appears.
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: 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)
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.