At one point the Alchemist is shown rinsing the hair of one of his victims. When the victim wakes up moments later, her hair is dry.
Neither federal law enforcement, intelligence agencies, nor their contractors have used simple strip shredders in nearly four decades. Modern government shredders all pulverize documents into dust and confetti that can neither be read nor reconstituted.
The subject profile brief on Tom Keen lists "Spoose's occupation" instead of "Spouse's."
In a common TV sound effect error, the scene in the kitchen when the guns are pointed at people a gun cocking sound is heard. Both guns used did not have external hammers, so they could not possibly make that sound. The sound effect is commonly used for dramatic purposes.
Insulin pumps are not cell phones. As such, there is no way one could be tracked as was shown.
When the agent reads the serial number for the insulin pump, at the end she says "Zed," indicating Z. The military/police phonetic alphabet term for Z is "Zulu."