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: 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)
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: 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: 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.
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.
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.