Character mistake: The actor who played FBI agent Ross is Lennie James, who is an English actor. When they are searching for the girl in the parking garage of her building, he orders the SWAT team and other agents to search everywhere, under the cars and to "check the boots." He says this twice. Boot is the British term for trunk. He should be telling them to check the trunks of the cars, not the boots.
Character mistake: When the EMTs are resuscitating Dr. Harris, they use a defibrillator despite their being on a wet metal dock. Not only is this highly dangerous to absolutely everyone present, it would be completely ineffective at restarting his heart as the water and the metal would dissipate the electric current too much for it to do any good. At the very least, they would have stabilized his spine then moved him to a dry surface before beginning defibrillation. (00:08:30 - 00:09:10)
Suggested correction: External defibrillators are self grounded and so can be safely used on wet or metal surfaces. https://danboater.org/travel-health-and-safety/are-aeds-safe-to-use-in-wet-environments.html.
AEDs are indeed safe to use in wet areas - but in the film, it is not an AED that is used. AEDs are automated, and not operated by humans, so as to reduce risk. Your link explicitly states the rescuer must not have direct contact with the body: they just apply the pads, then move back. In the film, it is shown in detail that the medic applies the paddles and then operates them while still in contact with the body. The mistake stands: it's a traditional defibrillator, thus incorrectly used.
Character mistake: When they deliver Xmas presents to Denmark it's already too late - Denmark has the presents on the 24th, early in the evening.
Character mistake: 2 men are watching the cameras with Bishop rolling away from the exploding truck. One man says to watch camera 2, but they are actually showing 1 in the upper left corner.
Character mistake: One of the latin chefs says in Spanish "Tengo un novia", when the correct way of saying it is "Tengo UNA novia". The way he said it sounds completely absurd.
Character mistake: Throughout the movie, the way characters pronounce Tori's last name changes. It switches back and forth from "Fredricking" and "Frederking".
Character mistake: A detective (Briggs?) told Sonny, " that I was stupid enough to believe your dark fantasy NINE-ELEVEN call." [emphasis added] An American law enforcement officer, in particular, should know and use "nine-one-one" (911) when referring to the nationwide emergency telephone number. "Nine-eleven" (written 9/11 and NOT 911) is used to refer to the September eleventh (9/11/01) terrorist attacks in the U.S. (01:15:57)
Character mistake: When they get the elephant, she is a she, but the boss man continually refers to her as "the bull", which means male. A female elephant is called a cow.
Character mistake: When trying to teach Banshee to fly, he is told to make his sound waves super sonic. Sound waves cannot be super sonic. Sound can only travel at the speed of sound, it always does. I guess they were trying to tell him to use a very high frequency, but the speed would still be the speed of sound.
Character mistake: In the pivotal scene where Bolly and Jonah are explaining the need to replace their departing players in the aggregate, Jonah adds up 3 players on base percentage and divides by three. You can't get an accurate average for a rate stat by that means unless all the players had the same number of plate appearances, which they did not. The answer given by Jonah Hill is 14 points low. When the older scout questions it, Beane scolds him that he "didn't carry the one." Same mistake is in the book.
Character mistake: Near the end of the movie, Dr. Bennett told Nathan that he could live with her and be able to finish the school year (without having to transfer). Assuming Nathan was a minor (under 18 years of age), Dr. Bennett did not have the legal authority to simply take custody of Nathan. There are proper court procedures to follow; a judge would have to agree and sign a court order declaring Dr. Bennett as Nathan's (temporary or permanent) legal guardian.