Continuity mistake: The door to Dexter's apartment opens out toward the walkway. However, in all subsequent episodes the door opens into his apartment.
Continuity mistake: Episode 3 begins with Dexter at the beach during a funeral. He is wearing a pair of stylish sunglasses. At the start of episode 4, Sgt Doakes is investigating a crime scene at the beach. He is wearing the same pair of sunglasses. Given the tense relationship between the two characters, it is unlikely they would share accessories.
Answer: No, Harry is not the real father. While he did have an affair with Dexter's biological mother (Laura Moser), this was when she was his C.I. Dexter was already born at that point.