This isn't modern family question. This is basic LAW and it sounds quite messed up and immature on your ex's part.
By law, if you have just cause, you can request to petition that your child NOT be allowed to spend the night with your ex husband if he is living with his girlfriend. You need a reason, not just b/c you are jealous that he is involved with another woman.
This is about the child's best interests.....not b/c you or your ex is upset that the other is involved with other people.
With that said, he can request the same of you. So if he does not want your daughter spending the night with you while your boyfriend stays the night and is living with you, it is fair game.
Lastly, where is YOUR lawyer??? If your ex has joint custody, he HAS TO MAINTAIN A RESIDENCE IN HIS NAME if he wants his daughter to stay with him on weekends, etc. If he does not have his own residence, than ABSOLUTELY, you can request that your daughter does not stay with him OVERNIGHT.
Your daughter is at an EXTREMELY vunerable age. She understands and is absorbing far more than you may think. Please dont bad mouth each other and let her know she is the top priority and is loved. Let her know she has NOTHING to do with this mess. Get her help if you can. Even though you don't think so, she is going through far more trauma than you know.
I have been divorced and I have 2 daughters. I speak with knowledge and best intentions. Divorce is hell for children and I am GREAT friends with my ex. The child's life is turned upside down forever and their schedules are never normal ever again.
Keep your fighting and anger away from her and please tell your ex to do the same. I've been divorced for 6 years, I still cry b/c my daughter leads a unstable schedule. Children first!!