"who else would cook, clean, and take care of baby while the other is at work..."
How about nearly every stay at home mom, and then some!
What exactly is he stressed about? Is it trying to balance the childcare and household needs while going to school? That's a tough load.
What might help is sitting down and making a plan for housekeeping jobs which feels equitable for both of you. Instead of each of you anticipating the other doing the housework, this will put it on paper and let each of you know what's what. Much better, from my experience, than doing it on the fly. I can't say you have a sense of "entitlement" per se, and from my experience as a Part Time working mom, couples seem to feel sometimes that the grass is definitely greener for their partner than it is for themselves. So, that said, sit down with your partner and figure out something that feels 'equitable for now'. Revisit the conversation in two weeks or a month, just to check in and see how it's going for both of you. Sometimes, treating a relationship as if it were a business partnership (and requiring that level of communication) is the best thing you can do. Less emotion, more exchange.
That said, if he's calling you ugly and telling you that no one would do this-- well, that's another sort of problem. Perhaps couples counseling? I don't think I'd do well with that sort of personal insult.