Wow. That is really low.
If I were this mum, I would absolutely take this as teaching moment with my kids. Very calmly I’d start with:
“Kids, let’s have a little sit down please. Your father sent me an email listing all the things about me that you don’t like. I’m not the least bit angry with you. I’m your mother and I want you to know that I love you no matter what. Like any human being I am not perfect and I understand there are things I could improve on, so let’s go through this list and talk it through. I want to try and be the best mother I can be.”
And really think about their points and come up with constructive ways of addressing them (if they are reasonable and true).
After that, I’d probably continue with “Right, can someone tell me why your father asked you to come up with this list in the first place? And do we think making such a list was a kind or useful thing to do?”
There’s no getting around the fact that he’s an asshole. But it might be an interesting exercise for the kids to think about whether this was a kind thing to do. I find most children, even from an early age, have a good sense of what is kind and fair and hopefully just by asking some probing questions they will think twice about engaging this kind of activity. Hopefully it will end with them coming to the conclusion that talking about others behind their back in a negative way is not kind and if the kids do have a particular issue with their mum they should bring it up with her directly. I would want my kids to feel safe telling me about anything.
Then, as another user suggested I’d end it with a “let’s say 3 nice things we like/appreciate” about everyone family member. Including the dad. I thought this suggestion was great.
I would then email the dad with the 3 nice things they said about their dad and say. “Thanks for your email. We did a similar exercise here but instead I asked them what good qualities they like about you. Here they are.” And leave it at that. Hopefully he will be ashamed of his behaviour. But if he comes back with anything snide just ignore it. You want to come across as faultless and whiter than white.
In the meantime, I’d definitely keep logs of all these kinds of behaviours. If they stack up you’ll want to share them with your lawyer / other professional.