Stop! Stop! Stop! Stop!
Although its very nice to see your concern for your children, they will survive not going on a few playdates for a while. These end for the foreseeable future. Communicate with the betrayed wife and coordinate it with her. Your wife and wankmuscle don't get a say in it.
If you do not impose any consequences for a WS they will not change their behavior. It sends a message showing that no matter how bad the betrayal - the BS will always take the WS back.
You need to change your behavior too. By not disclosing what was sent to your wife, she was openly disrespecting you. You need to shut that down. If she is unwilling to cooperate then you need to show that you are capable of leaving her and finding another woman who loves & respects you.
She slept with your friend and yet you have already jumped into MC without knowing the whole story. You need total disclosure with timeline and everything. What if they slept with each other in your bed? In your kids bed? etc. My intention is not to create unnecessary drama, but how can you forgive when you don't know what you're forgiving.
Demand a written timeline before you go any further. Ask her to leave the bedroom and move into a guestroom.