I'm really sorry for you all.
From personal experience, I couldn't eat when I was upset, anxious or sad, still can't as an adult. I barely ate after my dad died for about two weeks. My stomach was rumbling but I just couldn't eat, beyond cereal, milk etc. Anxiety and depression can cause sleep disturbance.
My friend told me about the daughter of her childminder having trouble eating, and they used to let her eat in front of the TV. It seems she could eat more while she wasn't entirely concentrating on the food, and it's quite widely accepted that people eat more this way.
If the relationship with her dad causes food refusal, it might be better in the short term if she eats separately from him, because any hint of discord or potential for a row is really counter-productive.
Get DD checked out at the GP, for intolerances/illness, and emotionally for anxiety/depression. It's possible that her dad has issues too - some just can't cope with their little girl growing up and becoming more independent, and feel they have to enforce their authority, including winning every argument.
I hope you can sort this out.