Feel the fear and do it anyway itsallmadness!
he has given his word and broken it twice. he wants to be a family, but only because he is worried about his reputation suffering. Well bully for him, if he cared that much, he'd have kept it in his pants.... and kept his promises. Make sure that he pays enough to maintain you and the DC in their education, get him to sign over the house if poss.
If in your culture divorce/separation is not accepted, I daresay chasing after a married man isn't particularly well looked at either, so if you have to tell someone who is blindly supporting her why you can't accept that, so be it. YOU did nothing wrong.
Cut FB from your life, stay in touch ONLY with those that are your number one fans, ditch, literally all the others. You can reopen it later with a new account when it is all done and dusted if you like.
This is the time to be brave, to show your DC how women/wives/mothers need to be treated.
You have given it your best shot, he didn't. Whatever happens, if you kick him out for what he has done, your family, the DC will look at you in admiration of your strength, and bravery. If you put up with this, think of the long term damage the situation will do to them, a mother desperately unhappy, but trapped, powerless to get out. That's not going to help you be the brilliant mum you are capable of being is it?
Sure it'll be down to you longer term to provide for your family, and he will have to be financially responsible, but you can and will do it.
Make it clear to anyone and everyone that the ex friend is a OW and will stop at nothing to steal bits of your life, and you are having none of it. If she wants to make her little points on FB, then let her. You need to have nothing to do with it.
Remember, cream always rises to the top!
Where are you, can anyone on here be of any help? Do you have any close RL friends to come with you to solicitors etc? Use whoever and whatever you need to, to get this situation resolved. You deserve better, your DC deserve better.