I mean, honestly, with a father like that it's not totally shocking she turned out to have some problems.
(I'm not talking about genetics or anything like that, I mean rather, that their upbringing was probably quite stressful for all of them, even if Rose was the "favourite". Not an excuse for being a petty bully, by itself... but it's easy to see where things might have gone wrong in, say, learning to evaluate one's self-worth and accomplishments. Or, just, learning to cope in unhealthy ways.)