See a lawyer and ensure you know your rights. It is just information, you need not act on it. However, knowing what H can and cannot legally do will be a relief for you.
Most MLCers are not big on responsibilities, and a D12, S10, and S9 are going to be a responsibility. Some run from the pressure and responsibilities of children. It is possible H is just manipulating things, threatening a schedule. To get you mad, or, and yes this does happen, or to give up later for more money in a separation arrangement.
How to deal with him. Do not accept any schedule he puts forward. He hasn’t even moved out yet. Ignore his ranting and talk to a L. Whatever you find out, do not tell H. Do not tell him you have seen a L. You need to protect you and the kids.
You were looking into purchasing a house. Did that happen?
As you said, this is the 7th time H is threatening to move out. Leave him to his sorting of clothes. He will leave or he won’t. Take the kids out for ice cream, and arcade games.
Treat him like a roommate who’s mad. Go out and enjoy yourself. Be dim/dark, business-like, and only about bills and kids.
Stay strong.
DnJ
Feelings are fleeting. Be better, not bitter. Love the person, forgive the sin.