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)
Revealing mistake: During the first major panic scene downtown, as the family get into the RV, one of the shots shows a man in a blue hoodie being 'changed' by the virus. He writhes on the ground in front of a taxi - the lettering on the door of the taxi is backwards, showing it's a reversed shot. (00:10:45)
Revealing mistake: When Gerry cuts the woman's hand off you can see it's an obvious fake arm and stump.
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.