I’m not sure the hate dominated, so much as was a necessary display. Think of it like Dumbledore in OOTP deliberately separating himself from Harry. There’s Snape the Deatheater, still publicly loyal to the cause, with Draco Malfoy in his house regularly feeding info to his father. Snape has to publicly hate Harry, any overt kindness to Harry blows his cover completely. Now I’m sure not all of the frustration and anger he leveled at Harry was acting, Harry is living proof of his unrequited love and frequently took glee in annoying Snape much like his father had, but I think the constance of vehemence of it was mostly cover.
Oh, I think Snape hated HP. HP was a constant reminder of losing Lily.......He plotted with LV to murder James and Harry to get Lily to himself......the fact that Harry lived must have torked Snape..........it was a constant reminder of the lost Lily.
I don’t think so. He was against Harry in private, as well.
He didn’t REALLY try to teach him Occlumency, and he was constantly putting him down to Dumbledore, too.
He had Lilly's eyes. Every time Snape looked at him he was reminded of his betrayal and loss.