The following are not legal advised.
Custody/Care and Control
1) Likely to be joint custody
2) Care and Control likely goes to you because the child is a female and he only comes home 2 times a week.
Divorce
1) You have 6 months to file the divorce based on adultery
Since he already have a woman outside, he will want out of this marriage too. Maybe you should speak with him and come out with ambicable outcome.
Don't serve on him yet because once he get his own, the battle royale will start.
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Family issue.
1) You can't force a person to admit it. Just take it as it is.
2) If shaming him infront of his family members makes you feel better then go ahead and do it. I don't see the purpose of it personally, is just personal vendette and not something I would want my kids to learn from me.
3) You may consider texting the woman, maybe the woman doesn't know he is married but keep it cordial.
With the above said, I hope both of you can talk things out for the welfare of the child. I have read cases of woman or man committing adultery, there is no right or wrong in general sense but the law allows the victim to file for divorce based on adultery.
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