Mike, sorry to here about this. It just seems like things are falling apart all around on both of our sitch.
Do not refinance the house. My W asked me about 2 months ago to do the same thing when the interest rates dropped. Now I know why she wanted to do it, she is going to try and keep our house if she is able to get enough money from me to pay for it.
You never know, I have heard many say that woman can not stand to be without their children, if you get 50/50, that just may change some things for you.
You stated she has been documenting things you say and emails you send, did this come out in the divorce papers?
Be tough and hold your ground on this, don't get suckered into going against what is best for you.