Other mistake: When the family's car is being hit by the ambulance, we see Gerry's head bouncing around by the impact in a close-up. Camera angle is from inside (approximately from the centre armrest). We see out his window, and there is nothing there. The camera angle changes to an exterior shot, and its clear that the position of the ambulance hitting the car should have made the ambulance visible from the interior camera. (00:08:05)
Other mistake: As Gerry enters the aircraft carrier command centre, we see screens with "world projections" of remaining uninfected populations by continent. They are counting down (illustrating the quick spread of the disease). However, the countdown speed of all of them, is approximately 3% each second - which means humankind would have no more than 20 seconds at this rate. The countdown speed should realistically be about 100,000 times slower than that (about 3% each day), which would still be the fastest pandemic spread the world has ever seen. (00:30:15)
Other mistake: When Gerry and Segen are in the plane towards Cardiff, a zombie attacks a flight attendant and a whole section of the plane. Despite the fact that what's happening would be obvious to the passengers of that section, no one even tries to escape the attack by going to the back of the plane, in Gerry's section. Gerry and the passenger have the time to pile up luggage without anybody from the front section interfering. (01:16:40)
Other mistake: Despite the number of staff and zombies (former staff) at the W.H.O Research Centre, the car park leading up to the main entrance is empty but for one car.
Other mistake: When Brad Pitt is on the plane - I think he's landing in Israel - he doesn't take his seat belt off, but that doesn't stop him from getting out of his seat.
Other mistake: When Jerry is in B wing, he has killed two zombies with his crowbar, but when we see the crowbar, there's no blood or brain matter on it.
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.