No, you're not being unreasonable in asking him to provide 2/3 of day care. However, you already know that he's not going to agree or if he does he'll not pay it. So....stop trying to get him to say OK. You're only adding more pressure for yourself and increasing your stress.
Don't expect him to agree but ask anyway and you might be surprised. What I'm saying is to expect that he won't agree so that you're not upset when he doesn't.
He's being unreasonable when he asks you to get a night and weekend job. In today's world we don't get to be that picky. You need a job, any job.
Or, it might be best for you to get state assistance until the baby starts school. So that you don't have child care expenses. It's not ideal but we do what we have to do. You can get child care and financial assistance as well as food stamps while going to school. You can get scholarships and grants to pay for schooling.
Definitely go back to court to let the judge know he's not paying alimony. He can be declared in contempt and be forced to pay. The state can take the amount out of his pay check. I don't understand why it's not already being taken out of his check. In Oregon, anyway, that's standard.
It's good to try to make a plan together but your divorce is new and it often takes time to heal before the couple can work together. I suggest you not rely on him for your plan. There is a light at the end of the tunnel and you have to step up and act on plans that you alone have made to reach it. Good if he co-operates but from his past performance you know he won't. You will eventually feel better once you accept that he's not going to help and you find ways to take care of yourself and your family without him.
Ask that all support be taken out of his pay check. This is the safest way to document what you get and don't get. Now, he can say, he's given it to you and you will have a difficult time proving that he didn't. And you'll get the money on time. No hassles. Get paid thru the state.