Plot hole: At the end the police shoot Cole while he was chasing Dr Peters but the police do not stop Dr Peters and question him as to why a man was chasing him at the airport with a gun, instead he is unchallenged and allowed to continue his journey.
Suggested correction: There is no connection between Cole and Dr. Peters, nobody knows he is specifically chasing him. Dr. Peters is beyond the gates and on his way into the plane so unless they want to stop the entire flight, which is unlikely, he is left alone. Perfectly reasonable.
Cole is chasing a man and Dr. Peters is running away (and knocking people over in the process). Further, they were screaming "there he is...stop him he's got a deadly virus" and pointing at Dr. Peters. Highly unlikely they would have let the plane depart.
There was a dude with a gun, a lot of people were running, panicking. Perfectly reasonable reaction from Peters. The police were after Cole anyway, so the attention was on him, not on Peters. And as I said, Peters was already past the inspection, so he was where he was supposed to be and there was no reason to stop him.
Not to mention that Dr. Railly can also corroborate the reason Dr. Peters is being chased.
Dr. Railly can corroborate the reason Cole is chasing Dr. Peters. When she points out that he is even on the front page of USA Today with a Nobel Prize winner, I think they'd definitely investigate.
The shrink was still alive and seemed to give up quickly on her mission to stop a virus from killing billions. Like she just let them take her while she remained silent instead of continuing to scream that he had a virus so Cole died in vain. And they would know why he was chasing them because all they would need to do was ask him.
Her concern was with Cole, she tries to protect him throughout the movie. After Cole is shot she knows there is no way they let her continue the chase, all she cares about is Cole. Cole is being chased by the cops, no reason for them to believe what he is yelling, no reason to believe her either, since she is as an accomplice.
Plot hole: How was Cole supposed to get on a plane without having any identification documents?
Suggested correction: Pretty easy to do pre-9/11.
I flew for most of my business career. Starting in the middle 80s, you had to have proper identification to get a boarding pass which let you on the airplane. each flight had a manifest with names and addresses of the passengers. Even with the post 9/11 changes, I think the current procedures are pretty sloppy compared to the 90s.
Revealing mistake: The paint in the "we did it" lettering in the roaring bear scene runs "downwards," according to the camera position, not according to the vertical of the shot buildings. (01:51:15)
Suggested correction: While yes there are slight angles involved, considering this was most likely a physical prop on-set, the direction of the camera tilt wouldn't be a factor. The direction the paint runs in doesn't seem wildly inconsistent with what would happen when painting something like that.