When Castle looks through the Pahntom file, the heading says NYFD followed by New York Fire Department. In reality, the New York City fire service is officially called the Fire Department of the City of New York and is abbreviated as FDNY.
When Ryan and Esposito exit the building with the rescue team, the building is still fully ablaze. Moments later, after Ryan sees Jenny and his baby, the camera pans up and there only on a few flames left on the ground level. The fire could not have been put out that quickly.
At some point near the end of filming this episode, Stana Katic injured her right hand. When Kate hugs Javier, her fore and middle fingers are taped together. The few scenes prior did not show this injury and no way could have held the gun a couple of minutes prior.