For me, I would agree to setting the date of separation earier, simply because delaying the date is delaying the invitable. It's better to get it over with, I think, and then move on with life.
He's definitely disreputable in his behavior. How can he collect SSI for the child without mentioning her mother?
Maybe you can't ask for some of that money. It's between him and the administering agency. However that income must be disclosed in the divorce paperwork.
Abused finances and irrelevant criminal histories are not usually a factor with custody sharing, but I think you can probably get primary custody.
Many lawyers will provide an initial consultations for free. I urge you to speak with some to understand the possibilities.