If any of those convictions involved harming a child, the father can get a restraining order that says the boyfriend can't be around the kids at all.
In some states, you can't bring the boyfriend/girlfriend around the kids until the divorce is final.
She's not going to get supervised visits ordered unless she can prove that the father is a danger to the kids. Sole custody is hard to get. More and more states go with joint legal custody, which means both parents have a say in the decisions.
As for who the mother consults with when you want to spend time with the kids on her parenting time, there's nothing you can do about that. She has no say, unless the kids are in harm's way, who spends time with the kids on the father's parenting time.