When Beckett is drinking coffee in the break room talking to Castle about how she let her mother down, in the close ups, she's holding the mug by the handle, but in the wide shots, she is holding it by the bottom.
When Montgomery calls Beckett into his office and offers her his "poor man's painkiller," you can see her open the flask and take a drink as the camera focuses past her on Montgomery. The camera then immediately changes angles to view her from the front, and she can be seen opening the flask again and drinking from it, though she had no time to lower her hand or close the flask.
It is mentioned repeatedly that no one knows what "Rathborne" looks like. But Coonan has already said he was in the service with Rathborne; U.S. Armed Services records include photos, which the NYPD can request.
Beckett cuts open a package of drugs and puts it to her tongue to find out what it is. No cop in their right mind would ever do this as it could be poisonous and is illegal. That's why most police carry or have access to drug testing kits.
After Coonan takes the gun from the police officer you can hear the sound of the gun being loaded(Racking the slide). At no time was Coonan's other hand near the gun to do this. Later when Coonan is confronted by Captain Roy Montgomery you can hear the racking of the slide again. Coonan's other hand was holding onto Castle therefor could not have racked the slide. It should also be noted that if the gun was loaded the first time we heard the slide rack there would be no need to load the gun a second time.