You are correct. This is a terrible situation for the children. Your fiance’s ex is more interested in making her miserable than in loving the children. It’s not at all good parenting to put kids in the middle of an old fight — which is exactly what he is doing. It’s unfortunate that the kids’ grandmother seems to be backing him up.
Sadly, there isn’t much you can do except to keep being yourself. As much as the kids may think they like the anarchy that exists at their Dad’s place, it isn’t healthy for them.
Do set clear rules at your place. Continue being kind but firm about how they are to treat you, their mother and each other.
There is no need to talk badly about their dad. That would only serve to keep the kids in the middle. Instead, focus on how things will be at your house. Don’t get involved in the fight about having them call you “dad”. Involve the children in coming up with a name for you that you all like.
The children may “hate’ you for giving them some structure and insisting on manners for the time being but they will come to appreciate you as they mature. Kids in situations like this usually do.
Meanwhile, I hope your fiance will have a talk with her divorce lawyer. If the kids’ father is hurting them psychologically, she may be able to get the divorce agreement modified so visits with their dad are supervised or more limited. Since the kids are having night terrors, the lawyer may advise that they be evaluated by a mental health counselor.
I wish you all well.