I joined this just to respond to your post!
I soooo feel you! My husband has custody of his two kids, who see their mother for a few weeks in the summer and a couple times a year, and our 11 y/o is ODD and anxiety, while his older sister is dx with adhd. We have two younger girls together.
The ODD is soooooo hard. I have experienced the lying, which just as you mentioned makes no sense!! The number of school conferences we had, and honestly I’m the one who is able to be there because my husbands work is not flexible. He destroys things too, like toys and snow boots (most recent find), throws things away that don’t belong to him, and weirdly sneaks food ( which is totally unrestricted in our house but he sneaks it into his closet. He has a clear vinyl lunch box and backpack after some issues with things he took to school, and also has a minimalistic room 🙂
I will say though that despite all this, I think we have a good relationship! What saved us is a book by Alan Kazdin called parenting the defiant child. The behavior stuff is not fixed exactly, but I feel like we can talk to each other and there’s some measure of peace in our home. He even told me recently about his first crush! If you’re not much of a reader or want a quick version, Alan Kazdin has a video series on Coursera, an app I put on my phone (all free!) and goes over all the parenting approaches there. I cannot recommend it highly enough. If you can get your daughters other parents to read/watch it too, I believe your child’s life could really change – and yours too, seriously. One thing I realize is that I can’t fix the hole in his heart and psyche, and my best efforts may be fruitless in the ways that I normally evaluate success, but if I can give our son a fighting chance of feeling worth something then he can have a chance! Best of luck balancing the impossible for your family.