Buy and read 'Emotional Blackmail' by Susan Forward. The way he forgot about your father's death, didn't even suspect that you dropping one of the balls might have had something to do with this, is hurtful in the extreme.
He is playing power games and it has to stop. He sees your attempts to start a discussion on the topic of making him an equal partner with each of you accountable to the other as a threat to his lofty position of Grand Poobah.
I note you are pretty sure he will sulk if you challenge him.
Presumably you have past experience of this. You need to persist regardless of his cold shoulder. If you want to discuss something he doesn't get to just fold his arms and say no and punish you for trying.
Write a list of all the items at home that you are responsible for - this will be a long list. Go on strike. Do nothing for him at home - absolutely nothing, and give non-responsive answers to 'Where's my...?' questions - until he gets over himself. Stop trying to keep all the balls in the air.
Rehearse answers to everyday questions like, 'Where's my water bottle?' - the answer is, 'Where you keep it.' He is not the lodger. He is a grown adult living in his own home, and is behaving like a spoiled child. Stop letting him get away with this.
While you're writing your list, research how much you should be paid for the responsibilities you have at work. My guess is you are underpaid. Bill him if so. Tell him you will find a job somewhere else unless he stumps up what you are worth.
His lack of engagement with the children is horrible and has probably already done some serious damage. Previous posters have commented on the effect of his treatment of your daughter, but if you have a son he is also soaking up all of this toxicity, observing the role women play.
You should find an individual therapist for yourself to delve into the relationship, the stress of it all, and to find support as you work to change it. It will be very hard to push him in the direction you want him to go. In his eyes you will be an uppity harridan when you set yourself to the campaign you need to wage and there will be pushback.
I suggest asking your children individually how they feel about dad and his disengagement from home, what they would like to see instead, what particularly angers or saddens them in their relationship with them and what saddens or angers them in their perception of your relationship with their father, or causes them shame. There is always shame in a family where there is a power imbalance. It's toxic.
Then you all need to do family therapy.
www.aft.org.uk/view/index.html?tzcheck=1
If he resists all of your efforts then I think you have a tough decision to make.