Ginger was killed primarily because he thought it was likely SHE may actually have been Sarah. Keep in mind at the time he had no photo to ID her with, unlike Reese. It wasnt until he searched the items in the apartment that he found a photo of Sarah and then knew what to look for. Ginger was a female in the residence associated with "Sarah Connor", thus a target. The bf was killed to prevent him from interfering more than likely.