Actually, we don't know what Lily's parents were like in canon, or how they treated their children, only that they were proud of Lily for having magic and Petunia resented the fact that they didn't regard Lily as a freak like she did.  Even so, that doesn't give Petunia and excuse in canon to treat Harry so horribly, like a servant instead of a family member. 

The nicknames the girls have in this story are reflective of personality, they aren't meant to be derogatory.  And while Hal might favor Lily more, which is a common failing among parents, my own do it, I wouldn't say that he hates his other daughter or wishes her dead, and he certainly never said that to her. He even says he regrets thinking that way, but he's hardly in a rational state after finding out his daughter is dying. After the disaster at the femily reunion, where both Vernon and Petunis were nasty withoput reason to both Severus, Harry, and Lily, Harry has no reason to like Petunia, considering she smacked him across the face, and then she insults his father right in front of him. Yes, he shouldn't have called her that, but he had just heard Sirius call her that, and was repeating what Sirius said.  Sev did call him on it, but considering what Petunia accused him of--making his wife sick and being an awful doctor, he wouldn't feel all that compelled to yell at his son for saying what he probably wanted to.  Petunia has never treated Severus with any kind of respect, either as a child or an adult, so therefore he doesn't feel she deserves any from him. 

Consider the conversation in the bedroom with Lily--not once does Petunia ever express any sort of sorrow that her sister might be dying, she asks after her halfheartedly and then proceeds to attack Severus and heap all the blame on him, calling him names and being generally nasty to a man who is falling apart watching the woman he loves die in front of his eyes.  And worse, she says such things to her sister, his wife, who obviously loves him, and whose days are numbered.  Whatever pranks Lily may have played upon her as a child, Lily still honors her relationship with her sister, allowing her into her home despite her behavior towards her husband and son and Petunia repays her by being her nasty vindictive self.

And there was never anything mentioned about favoring Harry over Dudley with Vi and Hal, simply that they see Harry more because he lives near to them.