It's a difficult one, I;ve never been in this situation, but leaving a yoghurt pot lying around doesn't seem to me to be serious - or is on a par with him leaving his own spoon on the sofa, which YOU then take into the kitchen. Don't do it, icemaiden - let him see that he can be casual about things too, nobody's perfect.
When they shout and scream, leave the house, go and stand outside the front door and let them see that it's unacceptable behaviour. Don't try and be middleman otherwise you'll always be piggy in the middle. He may be resenting your daughter growing up and developing her own mind - he has to learn that this is natural.
It sounds as if for various reasons your family runs in two halves - perhaps you could do like my neighbours do, they have a weekly meeting which everyone has to attend, to talk about who's doing what in the coming week, and what else needs doing. It works for them, they have to talk civilly to each other at least once a week.