They are saying that Joe Hill's birth certificate shows that his father was Joe Reagan. However, it was clearly stated previously by the mom that Joe was not told that he had a son and was not aware of the pregnancy. Therefore, the birth certificate could not show Joe Reagan being the father as you cannot put the father's name on the certificate without his knowledge and permission. The father has to sign an Acknowledgment of Paternity.
The grandfather's injury would have been reported to the NYPD via the NYFD when they treated and transported him.