Although I understand the feelings, naming and shaming is pointless. Name
and shame to whom? What difference does it make?
Legally,
unless she is a threat to you, you cannot kick her out of her own home.
What do you mean by "have it out?" Have a row? Have a fight? For
whose benefit?
As for the house - its a bit complicated
because I think you aren't married, so you'd need
legal advice about it. If its jointly owned,
she cannot sell it without your consent.
The most important
people in this are the children. I suggest if you focus on them you will
realise that naming and shaming their mum, or having a fight with her lover
isn't in their best interests. Making a fair split and staying amicable
over childcare arrangements is. It isn't easy, but it is the best you can
do for them.