mtb, I know the system [censored], but it is in place for a reason (so someone can't walk away with everything). Look at it as a fee to get her toxicity out of your and your kids lives. I assume you are going for full custody? That should help some with the financials.
I like OK's suggestion on the restraining order. I don't think there is much chance of it without any physical abuse. However, you do have a history with her and the police and they'll have reports on that that could help you, but I still think it will be a slim chance of approval.
I am supportive of you mentioning it to her about the false accusations. I would do it very business like, and not passive-aggressive. Something along the lines of "I've heard from a lot of people that you have been spreading false accusations about me. What is the purpose of trying to drag my name through the mud when you know I've done nothing to deserve any of this?" You can ask her to stop, but it isn't likely to work. Also, if you live in a state that allows for one party of a conversation to secretly record the other party I would do it. In my state you can. In states like California you can't, which is stupid. Make sure you are within your rights for that though since some of those wiretapping charges can be felonies.
mtb, as I said, you've put up with more than any other poster here. I know D stinks, and having to pay her to go away stinks. But one day you'll look back and realize you are better for all of this. You are still young. If I was 36 and my W cheated on me I would cut her loose like a bad habit! There are so many more options out there when you are still young.
M(53), W(54),D(19) M-23, T-25 Bomb Drop - Dec.23, 2017 Ring and Piecing since March 2018