Has he bought the ingredients and baked the cake yet?
That is the only thing that really counts here, from the perspective of the only person who really matters on their birthday - your five year-old. Does your DP not feel that?
As for the rest. Sounds complicated and like you're carrying him - and may both find it hard to switch out of that heiress / beneficiary, mother goddess / partial parent mindset when the time is right for that to happen, morally and practically (if it isn't already).
I do hope you're only living off part of the interest on your inheritance. Once capital is gone, it's gone and it goes very, very fast. With capital but low income, it's easy to misunderstand your situation as making you much richer and more morally bound to be beneficent that you actually are or can afford to be.
Great to hear you're training for something that will allow you to support yourself and your DC with earned income. I think you need to set yourself a more rigid timetable for studying and share that with him. Block out time for study and be firm about that. He is responsible for all DC at that time.
It's impossible to study while looking after children and running a house, anyone who has ever looked after children and a house knows that - which is why his questions about your studying ring loud alarm bells. They tell us he's domestically clueless and has always been carried and cosseted by women.
And/or he's a fantasist, of the 'it would be convenient to me if you could do all the domestic and childcare work, plus fit all your serious study hours into some magical night-time hinterland that allows you to study five hours a night, between 10 and 11pm and is only accessible to you.' (Obviously not to him, as time to do domestic chores).
Either way you need to make a firmer timetable and share out responsibility for tasks, so you're not leading on everything.