G.B.
He obviously has no parenting skills when it comes to discipline. Many other countries parent the same way he does though and they have a lot less issues with their children than Americans seem to. Some countries don't even tell their children "no" until they reach a certain age, such as 13 years old.
You are going to have to decide on some rules,no more than 3-4 to start with, and write them down and post them. Have a family meeting and talk about them and write consequences and rewards.
You are going to have to be more lenient and he is going to have to be a bit stricter. But if he and the ex-wife are on the same page you may have to realize it's out of your hands even if they live with you. They are the parents of those children and not you even though you are raising them, it seems he has told you that this is the style of parenting they have decided on.
These are some examples that might be open for discussion.
Our rules:
1. Pay attention when someone is talking to you, respond/reply in sentences.
1A. If you can't do this you are not able to go places and be around people. Privileges may be taken for several days.
2. Put trash in the trash can, if you don't it will still be there later and you'll still have to do it.
2A. If you do not do this then you will not be allowed to have similar items until you put the trash in the trash can. For example, no more cookies until the trash is put in the trash can.
3. You must have permission to go out of the yard.
3A. If you leave the yard then you will be grounded from going outside for the rest of the day and possibly up to a week. (Most severe consequences, if they are in the neighbors house playing for instance).
4. You must dress appropriately to go out of this house. Your hiney will be covered, your belly will be covered unless it's to gymnastics or tumbling. If it's cold outside you are required to wear a coat.
4A. You can stay home.