Even though Dodds' father told the EMT crew to apply a compress to the abdominal wound, and they should've done so immediately, when the gurney is being shoved into the ambulance, plenty of blood can be seen, but there is no sign of any bandaging or a compress to control the bleeding.
At Lincoln Hospital where Sgt. Dodds is in ICU, the only officers in the mostly empty hospital floor are the SVU unit, Captain Tucker and the elder Chief Dodd. When an NYPD officer is shot and recovering at a hospital, the entire floor would be flooded with cops showing their support for their wounded brother or sister.
During the early phase of the hostage scene, the pistol's safety is clearly on (in the down position), and it therefore could not have fired. But in the later close-ups of Munson's pistol, the safety has been released (safe lever is up) so he must've flipped it, even though that's very hard to do left-handed.
The bandage on Carisi's arm disappears awfully fast after he donates blood.