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)
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: 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)
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: 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)
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.