I don't think I'd send any photos to H. I could see him taking it two ways, and neither is good. He could think you were trying to pile guilt on him, or he could think getting a few photos makes him dad. Either way, it's not helpful.
Your mom.... well, you know that she thinks she has your best interests at heart. So try to remember that. And it's not like H hasn't given her reason to hate him. I think maybe you can tell her that you appreciate her support, and you understand why she thinks you should be doing more. But that you understand what needs to be done, and that you are taking care of it. Make sure that she knows that you are doing what needs to be done to protect yourself and the kids (you are, right?), and that everything else will happen in its time.
You are doing great! Hang in there!
On being friends.... there's a whole continuum of what the relationship with an X can look like. I don't think I will consider my X a friend, yet if I can do something to help her, I would. Mostly because it will be good for the kids. I don't like her very much, but I'm not interested in doing anything to hurt her, either.
