OP, haven't read the entire thread so apologies if you've covered this.
The thing that jumps out to me from your first post is that she sends your H letters, wanting dinner to discuss their DD, and contacts him constantly about things like hair cuts, etc.
This is your answer, she hasn't moved on. She doesn't need to meet your H to discuss their DD, email would suffice. Did he instigate the split?
Don't take her behaviour personally, I think she would be like this with any new lady in his life, because she isn't ready to let go. That's her problem though, not yours.
Your H needs to sort this, by continuing to co-parent but to firmly show her that his life has changed, and her changing her name is completely irrelevant, and doesn't even warrant comment. And you must do the same.
She is right, they are and always will be DDs parents. But if she's happy for you to do the lion's share of co-parenting her DD, she must acknowledge the role you play, and your H must support you and let any guilt go about the past. You're the adults, you need to take the emotions out of it, and stop reacting to her.
Your lives have moved on, her's is stuck, and she may be clinging on to the past. Or a bit jealous of your lives, so trying to wind you up. That's fine, because if you simply ignore her attempts to do this, you'r problem will go away.