Raymond Vance claimed that the name of his client was privileged, but he had committed a crime to help that client to commit a crime. This is conspiracy, privilege no longer applies.
Vance claims his client's name is privileged information. This would be true only if he revealed privileged communication between lawyer and client by identifying the client. The client's name by itself is only confidential.
When the suspect, Emma, is wrestling with a hostage for her gun, Castle attempts to grab the gun. The gun is pointed at Castle lower chest area (just below 'Writer') when it goes off. However, even though it was moving in an upward direction, it still wouldn't have been high enough when it discharged to have hit about the 'Writer' printed on his vest.
At approximately five minutes in, there is an outdoor (street) shot which was obviously shot in L.A. However, in the shot were NYPD officers, NYPD police vehicles and, looking very closely, a LAPD black and white. There should not be any LAPD police cruisers in "New York."
When Beckett is bemoaning losing her job with the government, she comments "I can't believe the FBI fired me." But she worked directly for the Attorney General; and although the AG's office and FBI are both under the jurisdiction of the Department of Justice, it's implied when Beckett began working there that the AG's office and the FBI are separate agencies.
Near the beginning, when Beckett is complaining about getting fired from her government job, she says she hasn't been out of work since she was fifteen. But in Always (2012), she is suspended by Captain Gates and chooses instead to resign from the NYPD. And, when she wants to return at the end of After the Storm (2012), she is required to serve out the rest of the suspension.