When Beckett is searching for the suspect you can hear the sound of a shotgun being loaded. Beckett turns to find the suspect is standing behind her holding a pump action shotgun. After a few words are exchanged the suspect cycles the pump action shotgun again presumably to reload but since he never fired it there was no need. There was also no ejection of a shotgun shell that would lead you to the assumption there was never any ammo in the gun.
When Mr Smith opens the envelope from Captain Montgomery, it includes a newspaper clipping of Detective Beckett getting shot. That event happened at Montgomery's funeral so he could not have sent that article. CORRECTION: Mr Smith opens the envelope and slides out its contents on top of the newspaper, it was not an article inside the envelope. Mr Smith got the files after Beckett was shot, which Castle explains to Martha (32 min).
When Kate corners the guy trying to get out of the fire escape and he turns on her with gun extended, we hear the sound of an automatic pistol being cocked, although he is holding a revolver.
The envelope to Mr. Smith gives Montgomery's address as 299 First Avenue, which is between 17th and 18th Street. Earlier, Montgomery was shown to be living in a private home with a lawn, which does not exist in the area of the given address.
It is unknown why would a shotgun armed killer put his gun down in a standoff vs a cop armed with a 9mm pistol. Her bulletproof vest would not protect against a 12 gauge point blank. Also it does not cover her head. so the character motivation of the scene is unclear. It's all "put your gun down" against "no _you_ put your gun down" - a classic Mexican standoff. (Actually, a Level IIIa vest all stop 12 gauge buckshot and slugs, although she would probably have broken ribs.)