Do you think your DD does actually hate your DP? How soon did he move in after you split from your ex?
Does she resent the baby do you think, and takes it out on your DP as she knows she'll get in trouble for venting at the baby?
Your DD could be really struggling with the changes she's had to endure and the upheaval of being away from you while the baby and DP get to stay there (even though I'm sure she enjoys being with ex too).
You may just need to really reassure her and make sure she feels that she can tell you about any feelings she has, good or bad, without getting into trouble.
WRT to your DP, we all have different ways of parenting and there are bound to be some elements of discipline that your DP has already dealt with in his grown up DCs that you are still finding your way with your own DD, so you will probably evolve ways to deal with situations together as your DD grows up.
My DP and I have fairly similar boundaries and outlooks on childcare in general but I know that if we'd parented from the start that I would have done lots of things very differently from the way he and his ex did and vice versa.
Even now, there are times when he thinks my DCs are a bit rude to him or to me, there are times when I think the way he deals with his own DDs is ineffective and too soft, but as we only have to do this together once a week and during holidays I have learned to let it be rather than get wound up about it.
If we lived together full time we would have to sit down and work out what our house rules are, what the consequences for are for not adhering to them etc. If your DP is having to be a father figure when your DD is in your home then he needs to have some say in how she behaves, so I think you need to sit down together with your DD and chat about the sort of home you all want this to be. Get her involved, kids are happy to be part of the decision making process and enjoy coming up with rules and rewards.
It is hard having to parent someone else's children, especially when they come with an attitude! There isn't that unconditional love that you have for your own DCs to carry you through the tricky stuff, so give your DP some credit for doing a hard job and try to find a way to make it more harmonious for you all. It's in your best interests not to brush this under the carpet - if you read the threads on this board there are so many people who say that if they had their time again they wouldn't get into a relationship with someone who already had children, it causes so many problems. Try to nip your problems in the bud, as they won't go away on their own, they'll just fester and cause resentment from either DD, DP or both!