OK. from the snapshot you've given, and let's face it - it is only a snapshot - I'd say your relationship is in trouble. Do you want to put the effort in to continue it? Do you and your partner have enough love, trust and respect to put in the work to get it back on foot?
Was there ever a time it was good? What I'm going on to say, is premissed on a "yes" to that. If the answer is "no" - that's a whole other thing.
You are both stressed. you are both working really hard outside the home. Relationships that survive most often stay the course not because they don't encounter difficulties but because the partners negotiate those stresses to their satisfaction. At the moment, you two seem to be at war.
He is responding to your stress (and his own) with belittling of your work out of the home, disengagement and (it seems) with a bit of a house-work strike.
You are responding by belittling his parenting and disengagement.
To be honest, when things have reached this state you probably need to see a counsellor/outside third party, to a. acknowledge you are both still committed to the relationship b. create a neutral place of discussion c. sort out some ground-rules for polite, meaningful communication that is directed towards problem-solving rather than retaliation.
I am extremely angry on your behalf about the undermining of your paid work, and that he is adding to your work-load in the home. But I think I'd be looking to find out if that can be sorted, if he is prepared to do more, if there are ways for everybody to do less, rather than going for this war of attrition.
That way, at least you find out fast. If he won't try ... well, that tells its own story. But if there's more to it than that - you'll find that out too.
It may well be that he is genuinely, fundamentally, contemptuous of you, and won't/can't change. But at least you'll know.
Perhaps you might find out that it's less bleak than that and manage to get things back.
Wrt his parenting. If he is genuinely less competent than a childminder, then you stop him looking after the child on his own. But you cannot then be angry about him not looking after your daughter. If he is competent enough, then don't chip away at that - it is totally self-defeating, and against your long-term interests. And hers.
Wars of attrition are protracted and exhausting. It may well be that he is not doing enough/cannot be/do what you need/want. But it really is better to grasp that nettle and find out, quickly, rather than drag a dying relationship around with you for years.