OP I bet you're wishing you'd never mentioned it now 
Reading between the lines, rather than commenting on how you have tried to explain your feelings, I think it's likely that you're struggling with the responsibility of everything and your OH is distant and possibly acting in a passive aggressive manner. You need to be communicating your concerns with him to see how he reacts when he knows you are so upset and fed up with it all.
I don't see how you can realistically be studying if you have three under twos and no help with them, let alone keep a lid on everything else if you are also responsibly for all the admin and house work and intend to be working part time. I can understand your frustration because it is natural to want to improve your life, but would suggest you defer any studying until your children are a little older for the sake of your own sanity. I'd be focussing on whether your OH is the OH for you.
I imagine that your family are already aware that you aren't happy as they probably know you are ambitious. Their support and employment of your OH are probably continued as they can see you and your OH are different kinds of people and they are trying to help keep your world afloat rather than see you suffer. Ask them for help or advice and also see if you can have the money they gift you put aside incase you need it for yourself and your children at a later date so it isn't spent on coffee machines.
For what it's worth, and having previously lived with a responsibility shirking OH for many years, I would say that if he is not making any effort now it's unlikely he ever will. Make sure he does actually do the things he is responsible for because the last thing you need is to find he hasn't done something important. There will always be an excuse he can present you with, but actually not wanting to do something is a good enough reason for him not doing it, for him anyway. Everyone has free will. Keep an eye on things so nothing can go wrong.
I'd be far more concerned that he had no empathy for you when you were exhausted. At best he could have let you sleep. You could even have both stayed awake and spent the miserable time together! A partner who always puts themselves first is not a partner.
If you think he is both callous in his attitude to you and not capable of taking full responsibility for himself or his obligations to his family (and that is not necessarily about money) then I would be examining your options in more detail before talking with him. Start with a frank but careful discussion of the number of hours you need to sleep and the support you will need. Things are more easily sorted out when you are sleeping enough.
Good luck 