I understand your concerns. When I met my husband, he was ending a marriage with 2 kids. I didn't meet them for a while, and I didn't stay over when they were there. My husband shared an apartment with another divorced guy, and they had their kids on opposite weekends, so that's where the kids went. Later my husband and I got a place together, but I moved out on weekends (hiding my stuff in the attic and staying with friends).
I'm going to urge you not to get involved with the girlfriend. Don't meet with her. She's not the enemy, she's not the parent, and you have no control over her at all. At worst, you will make things more complicated and empower her, as if she has some say in your child's life. Even if things go perfectly, you are telling your ex that parenting is a "woman thing" and he's not a factor!
The fact is, your problem and your challenge is with your ex. So you are together until your child is 18 and you have to learn to co-parent. Your conversations MUST be with him, or with his attorney through your attorney. Period. That doesn't mean you are rejecting her involvement at some point, but that she's not a parent in this situation. It's not up to you to advise her, correct her, direct her or encourage her. If she stays in the picture, fine. If their relationship fails, then you really don't want to be invested in her.
You absolutely must work with an attorney or at least a mediator if your ex agrees and you think you can work these things out with just one legal representative. Your attorney will recommend a custody/visitation agreement that should include overnights/contacts with new partners/dates for both of you. Don't make it about HIS dates - make it about BOTH of you so it's equal. Both of you can agree to only be with another person if the child is not there. If your ex wants to see your son for overnights, he can take him to a motel, to Grandma's, to Aunt Susie's, to a campground, or his girlfriend can do what I did - move out without leaving a trace. (For me, furniture was fine - the kid can't tell - but no clothes, no accessories and no toiletries.) If it's too much of a hassle, then he's not that devoted a father, and he can see his son during the day.
Get past the idea that you get to decide if he's in good hands if you're not there. You must, as a woman who has been denied equal rights (as we all have), be willing to extend equal rights of parenting to your husband. You cannot pass this off to the woman just because she's female. That's actually telling your husband that he doesn't have to do the parenting. It's sexist, and it's a huge mistake long range. He's the dad. He's going to make beds and make lunches and do laundry and fix boo-boos and pack clothes (or buy the correct sizes) and stop handing everything over to a woman - you or his girlfriend. He wants an open relationship? Fine, he's got one, and all that it entails.
And DON'T give him visitation in your home! He's out, he's out! He can manage the inconveniences! Don't help him have less responsibility! Besides, it will confuse your son and make him think Daddy is coming home. Daddy lives elsewhere - that is the reality of divorce. Don't muddy those waters.