I'm sorry to hear that your daughter is in this position and that it is affecting you so greatly. You're a great mom to be standing beside her through this and being such a wonderful support.
Regarding the labour room, even if the father could somehow prove his paternity before the baby was born, he has no right to be in the labour room unless your daughter chooses to have him in there. Most hospitals will only allow one or 2 coaches or visitors in the room at a time, and if someone upsets the mother they will bar that person from the labour room. This includes spouses, much less boyfriends and ex-boyfriends. Just make sure that the hospital staff is aware that this guy is toxic to your daughter and that you want him and his family as far from her while she is going through what's already one of the most difficult (though miraculous) things a woman (much less girl) can go through. She may have to inform them of this herself, depending on the hospital.
Regarding the apparently toxic father and his family, I hope you are recording the hurtful and abusive things he and his family are saying. These comments may assist you in the visitation/custody procedings following a positive dna test. Get a notebook and write down the comments, along with the time, date and place they occurred. I am not certain they will help, but I can't see how they would hurt your case.
I can totally understand your daughter not wanting her baby going to their house unsupervised. I wouldn't want my child in the sole care of these people for so much as 5 minutes either.
My heart goes out to both of you during this time.
Hoping strength for both of you.
Jacqueline