When Katharine James fires her gun in the restaurant, the sound and muzzle flash match, and the trigger and hammer move, but unlike a with real gun the slide stays still and no shell casings are ejected.
Paramedics and EMT's rarely if ever administer medications like antibiotics and analgesics orally. Mainly because oral medications take too long to work, paramedics treat acute injury and illness, which require the use of subcutaneous or intravenous medication due to them having a much faster onset of action.
During the briefing on Patrick Cross, he says that Cross is 34, and the information displayed on the one of the screens show his birthdate as September 26, 1984. Then Kristen says that Cross witnessed his father kill his mother and sisters when he was 10. However, the newspaper articles on the screen reporting about the murders are dated January 14, 1991. Based on his birth date, he would have only been 6 at the time of the murders.
Deputy Inspector Talmage is wearing the wrong style of badge. In the NYPD a Deputy Inspector wears the same style badge as a Captain, he is wearing the style of badge worn by chiefs and commissioners. Plus his badge has four stars on it, only the Chief of the Department and the First Deputy Commissioner have four stars.