Stupidity: They say the airline industry was the perfect way to transmit the virus but how would the plane even fly if there was an infected person. In one scene the plane crashed within a minute of an infected person being on it.
Stupidity: Throughout the movie, and especially up on the top of the apartment building before they get rescued at the beginning of the movie, Gerry counts 12 seconds to check if someone has been infected. Just to mention this apartment example, he gets blood on his face and spends more than 12 seconds getting up and to the edge of the roof, plenty of time to become "Zeke." He then counts 12 seconds, as if they would actually matter to his infection.
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.