Other mistake: Several times during the film Captain Ramey's police badge is displayed on his belt and it is a NYPD Detective's badge, not a NYPD Captain's badge. The two badges look very different, so they are easy to tell apart. (00:35:10)
Other mistake: In one scene we see a "chalk outline" on the ground of where the pimp's body was laying. This is something that is never done in real life given it can contaminate the crime scene. (00:34:00)
Other mistake: Towards the end of the movie, after the sniper talks to Stu in the ambulance, the camera rises, and you can see that the street and sidewalk are wet, and below the police cars, the concrete is dry; when in the whole movie, it never rained at all.
Answer: The pizza man is intended by the Caller to be a decoy so he can get away. The Caller sends the pizza man to the phone booth so that Stu will identify him later in the apartment, giving the police a convenient answer to the mystery while the real Caller escapes. In fact, at the end of the movie, the Caller comes to visit Stu - obviously he's not the dead pizza delivery man.
Phoenix