GF, Your D is only 12. As her mother and it appears her sole protector you need to set some boundaries about her visitations with her father. Apparently he is not concerned as to what she finds when she visits him. Document these things and tell him that if he continues to have these types of things where she can see them that he will have to visit her at your place or take her out. It is definitely not a healthy environment for a 12 year old. When you set the boundaries be firm and in control. Do not let him push your buttons, but make him understand that what he has in plain view is inappropriate for your daughter. MLC or not, NO one messes with mama bear's babies!
As far as a MLC lying believe nothing they say and only half of what they do. Please prepare yourself that there is or may have been an ow. At the very least if you continue to be intimate with him protect yourself.
You know your H better than anyone. Does he look like he's truly happy? My H tries to make it look like he's done the right thing by leaving and living with ow but my D's call it "Fake daddy happy." My H's eyes are hollow and his skin is gray. He's definitely depressed and unhappy. Sooner or later most wake up and realize that you and the M weren't the source of their unhappiness. Until the time that they are willing to face their issues and deal with them, they'll keep running.
Has your H asked you what you think of him dating?
Hang in there GF. Patience for the LBS is the key.