When Katie is holding her baby, the blanket changes between covering his head (with her hand on the blanket) and being down lower, leaving his head exposed (with her hand on his head).
When J.J. threw the syringe at Dr. King, the plunger didn't get depressed, so it wouldn't have been injected into him.
J.J. calculates the trajectory to throw the syringe at Dr. King, but then throws it on a different path.