I think he was just as much a father figure looking out for Harry as Remus, Sirius, Dumbledore and Hagrid were, except he was seeing James and remembering all the bad times he went through, maybe even taking his regrets out on Harry for losing Lily to James. But he was always there. Maybe only because he was Lily's son, but Harry needed someone to be strict with him, if he was to go against Voldemort one day. He was mean for no reason yes, but that was repressed memories of James IMO.

I agree with everything you said, plus I want to add some more. Harry hated Snape indeed, but although he would probably not admit, he always respected Snape. He recognised him as an extraordinary wizard and he empathized more when he found out that James mistreated him. He used to suspect Snape about everything bad that happened, but when Snape actually killed Dumbledore Harry was completely taken aback, as everyone else. So, deep inside, even Harry trusted him. On the other hand, Snape cared about Harry just for Lily's sake, but he was cruel to him, because he hated James. I think there is a lot of depth here. Everyone but Snape always said that Harry is more like his mother as a character, while Snape said he was just like his father. But think about it, Snape is perhaps the most intelligent character besides Dumbledore (and Voldemort?)! There is NO WAY he didn't see the character resemblance Harry and Lily shared!!! Therefore, I think he was so cold towards Harry in order to protect himself from another deep emotion. He was prepared to see James' kid from the very first moment, not Lily's. Also, Harry was always kind, honest and good to everyone, except Snape, to whom he was really rebellious. He showed a James attitude towards Snape, which is something Snape would never forgive. There was a strong bond between them, of a really rare and complicated nature. And it wasn't plain hate!

^ In short, Snape perhaps even loved Harry; he just never let anyone - not even himself- realize it.

A father figure would be someone whom Harry looked up to as a father, or looked to for fatherly guidance and a fatherly role in his life. Harry never looked to Snape in that way, even after Snape's death when he knew the truth. He respected Snape but didn't see him as any father figure.

Yes, he worked behind the scenes to protect Harry, but he was in no way a father figure to him. I agree with Flickerflame.

They still hated each other, even though Snape did protect him

Well I think honestly we will never know how Snape felt about Harry-but I think that he did care about him in a way towards the end. Snape is the most complex character, that is why it is hard to say. A father figure can be someone who protects you, you dont have to always look up to that person.

But you have to see them as a father. It doesn't matter if they see you as a son or child, or how they regard you at all. It doesn't matter what Snape thought of Harry, and we know exactly how Harry saw Snape since the books are almost entirely from his POV.

I think there is a difference between a father-figure and an older man who tries to protect a younger person. Harry didn't consider Snape as a father and Snape didn't consider Harry as a son.

Of course not, Harry never regarded Snape as a father figure.