It's not really about the minor things at all - they just present opportunities.
Agree that the counselling idea won't work unless he acknowledges first that there is a problem and he owns it. Maybe he is so habituated to it that he can't see what he's doing at all.
However, perhaps he is unconsciously exploiting the fact that they are in your house to make her feel worse than ever - it's a lever.
Have you tried Harka's suggestion about confronting him yourself and pointing out his behaviour to him? This is more likely to work than your daughter doing it.
Beyond that I would say your daughter needs to stop rewarding his bad behaviour by paying attention to him, or appeasing him - but to stay centred in her own happy self and go off and do her own thing. This is not easy and takes some determined effort.
I suppose the flip side is also to reward him when he manages to deal with 'stuff' in a balanced, adult manner.
Ultimately and inevitably, if he continues to behave like a child your daughter will lose respect for him terminally and will not want to stay with him (whether she realises that yet herself or not). If I were her mother, that's what I'd tell him. I'd tell him my daughter wants a man for a partner, not a child. I'd shame him with some raw home truths, but I'd do it in private.