Hi M.,
I don't have any experience in this type of situation, so my advice may or may not be helpful. So here it goes..........
I think if I was in your shoes, I would start off with never ever saying anything bad about the mother of your man's child. At least not to your man. I would stay neutral and supportive.
Vent to your girlfriends as to what a pain in the neck she is.
As annoying and abnoxious as the child's mother sounds; you need to remember that your man will always have a bond with her, and she will always be around. Be proud of him for wanting to be involved in his child's life. And part of being involved is having to put up with all the foolishness and bratty actions of his ex. Remember, there is a reason that he is no longer with her.
As far as just walking into your home like she lives there. You definetly need to establish some boundries. That would drive me crazy also. I personally would keep my door locked if I knew she would be coming over so that she would be forced to knock or ring the door bell. I would think that eventually she would get the message. Always greet her at the door with a welcome smile and greeting. A genuine smile. Not sarcastic. She will be much easier to tame with smiles and niceness, even if inside you are growling like a bear. Good Luck with your situation. I hope I added some helpful insight.