Corinne3, your husband's behavior, from what you've told us on here, has been abominable. Why would you want to accept that behavior back into your life? Accepting him as a friend on FB sends him the message that "I want you back in my life." You may* think* it's because you'll be a good influence on him etc...but the only message he'll receive from you adding him as a friend on FB is: "Yeah...she still wants me."
I understand that you want to change him. But you cannot change him. Only your husband can change his behavior.
It's obvious he has zero self-esteem. Only people with low self-esteem and zero respect for themselves cheat, lie, and play mind-games. They do this because in a way, they hate themselves...and it's a sad thing to witness. A person who respects himself would never defile his honor in such a way.
So...he seeks out others who will confirm this low opinion he has of himself...other women who cheat, lie, etc..
Like attracts like.
It's as simple as that.
You know what the irony in all of this is? It's that when you* finally are able to shut the door on his behavior...that's when he'll come crawling back. Because he *wants rejection. He seeks it out. Again, this is a symptom of his low self-esteem. He runs from you now, because he sees how undeserving of you he is. Ahhhh! But do not feel sorry for him!!! Do not feel sorry for his weakness. He chose this path. And he must choose otherwise, or risk falling deeper into the well.
And again, please understand this one thing - you cannot - you cannot - change him.
He must change on his own.
If it helps you to stay away from him, why not try placing that challenge before him? See if he can rise to it. See if he can make himself worthy of your respect once again. And in the meantime, have enough respect for yourself not to take him back unless he does.
You must be strong - for both your children and your own sake. Someone has to set an example for the kids...and unfortunately, so far, it looks like you'll have to be doing it alone.