When Castle gets to his apartment he opens the door and walks inside away from the door. The door is stopped in the full open position. The camera angle switches to the front view where Castle continues to walk inside but the door is now closed. At no time did Castle attempt to close the door nor was there any type of auto closing device on the door.
Lucy the female dog, who is called "a good girl" in more than one scene, is clearly a male dog in the final scene.
While trying to find the amnesiac's identity, the detectives found an asthma inhaler in his pocket. If this were a prescription medication, this would have had the patient's name on it as well as the information for the pharmacy and the prescribing doctor, but the inhaler shown looks very much like Primatene Mist, which is an over-the-counter asthma inhaler.
As a prescription medication, the inhaler "should" have the patient's, prescriber's, and pharmacy's information. However, in real life, the pharmacy usually puts the label on the box, not the actual inhaler.
In the witness statement that Beckett showed Darius (the victim's assistant) the first line says "On the afternoon of January 12, 2010..." The original air date of The Fifth Bullet was December 7, 2009.