OP. I’m so sorry to read about all this, but, can I ask, why did you have to pay for the cleaner and babysitter out of money your parents gave you? This is a joint expense - does your DH not see that?
I really sympathise with a lot of what you’re saying because I’ve been there myself. There was a time when I had 4 DC under 7 and a workaholic DH. To be honest, it was easier when DH was away with work (which he generally was for at least a couple of nights every fortnight) because that was my “break” - ie not having to make an effort with dinner, tidy up before he came in, etc. Also, not have to listen to his criticisms and “have you done x?” “Can you do y?” on top of everything else. When you described your DH’s habit of making criticisms as a way of managing his own stress, I can totally relate to that. And yes, if you answer back or snap, you are told you’re being “difficult” or “snappy”. At least you’ve recognised this dynamic early on though. It took me about 14 years to see what was going on and I only really verbalised it in the last year or so since I’ve been in therapy. Since then I’ve been taking small steps and it is starting to make a difference.
The only thing I would say to where my DH sounds different to yours however, was that he didn’t mind at all how many hours I booked a cleaner. I did have a bit of help on and off, but with 4 DC it never really meant much of a break anyway tbh and sometimes the “help” was just another thing to manage. DH never begrudged me that though, I have to say. I had no family around, but loads of friends and we live in central London so loads to do and easy to get out every day etc. I think if I’d been somewhere rural, I wouldn’t have coped. Also, to be fair to DH, he never expected me to go back to work; no pressure there whatsoever and I never had to account for anything I spent on myself or the DC. He’s very easy-going and generous in that sense.
I think if my DH expected me to do all that you do and then work / study in the evenings, that would be the final straw. You need to lay it on the line for him.
As for reading blogs about women who seem organised etc - WTF?? I would throw the bloody laptop at him.
Tell him in no uncertain terms - a) you are exhausted b) if you end up having a breakdown you will be no use to him or anyone c) you cannot cope with one more criticism. Not one more. Even if he doesn’t mean it as criticism, this is how you hear it and it is making you ill d) go to the GP e) tell him you need a regular break in order to function as a good enough parent to your children and if he doesn’t sort this as a matter of priority, you will take yourself away for a weekend and he can see how he gets in alone. This has hi to the point now where you have to protect your sanity. Maybe when he is left alone with the kids, he could consult some blogs about “do it all dads?” That’s bound to sort it 
Also let him know in no uncertain terms that if he wants the “do it all” housewife / mother as his wife who never complains, then he needs to be the “do it all” sole financial provider who gets on with that role and never complains and /or restricts his wife’s access to money or expects her to take on anything else outside the home. He can’t have his cake and eat it frankly.
I wish you all the best and take it easy on yourself.