Hi there, KiwiGal - I was asked to stop by here by JellyB to see if I had any suggestions to your situation. I have a lot of experience with blended families and working with difficult situations, including co-parenting with mentally ill and abusive partners.
My initial thought about your boys is that they'll be fine. However inappropriate this is and however flaky their dad is, they are old enough to not be too impacted by this and they will find their voices in the relationship with him. I don't see any signs in what you wrote of him being a very destructive influence on them.
Remember, he is not the same person towards them as he is to you (albeit still acting flaky). They have a different dynamics. Very shortly, they will get to the age where friends are more important than parents. If I were you, I'd focus on having them around male role models who are reliable, trustworthy and accomplished.
Also, OW could potentially be a positive presence in their lives. I know that's hard to wrap your head around, but if she remains in H's life and is nice to your boys, she could be a champion for them in relation to their father. I don't think her presence - although they would prefer to be alone with their dad - is going to impact them negatively. The only clever thing you could do, is perhaps tell him that forcing her on them has the opposite effect of what he thinks - they dislike her more the less time alone they have with him.
It sounds like he comes from a family of origin where they sweep things very thoroughly under the rug. It's hard to fathom grandparents who stop having contact with their grandchildren. I'm so glad you have been able to create a good existence for you and your children. You should be very proud!
I hope the financial settlement will be over soon. Have you thought about filing a police report over the other missing items? Tell XH you have to in order to get an insurance claim?
Best wishes!
M 16 yrs, WH62, P54 3 adult blended kids EA 11/13, BD1 6/14 PA fall 14, BD2 2/15 Piecing 2015, BD3 12/15 Separated 4/16 WH moved OW in 5/16 Divorced 6/15/17