OP - i think your husband is actually showing signs of being a controlling and abusive person.
He seems determined to be in control of ALL money that 'comes in' and it seems that so far HE has been having the power of veto over financial decisions - simply because it's his wage/money.
I can see red flags re financial abuse waving.
Talk about putting things into the family pot - I put a baby and two years of my life in there....We chose to have a baby so by extension we chose to pay the expenses of having a baby. Whether that’s nappies, clothes or repairing the damage the baby caused
Precisely.
Any health problems - be they mental or physical - resulting from having dc and rectifying them goes into this pot.
You gave up your job to be a SAHM so re-training for a child-friendly future job/career is also goes into this pot.
You sacrificed your own personal financial freedom and autonomy in order to accommodate a 'joint' decision to have dc.
HE KNOWS that him being the only breadwinner/billpayer is his contribution/sacrifice towards the family pot.
He also knows that 'his' wage is actually yours too by virtue of the commitments/decisions you've both made.
Yet he CHOOSES not to see it this way.
He wants it all his way, he wants all the power over 'joint' decisions and he wants you to put up and shut up 
Re the car - he had one to travel to work and was happy to let you take public transport despite living rural and being heavily pregnant.
At what point did your husband make your and your future dc's transport needs and safety a priority?
There was 'joint' money available to afford a baby - where was the money for a car?
Let me guess, HE decided that 'we can't afford it/you don't need it'?
So your dad stepped up and provided for your needs - out of his own personal money.
The petrol that you buy is bought with YOUR 'wages'.
I’d just like to have a job with prospects instead of a zero hour contract. I haven’t thought about it because I couldn’t afford it...He’s offered a specific amount with a suggestion that it would cover what I need to get my life back and stop crying all the time
How does he knows the 'specific amount' that you will require when neither of you know precisely what it is you want to do? Or if it's even possible?
Have you discussed him stepping up re housework, childcare and life admin for when you go back to work?
Or pay arrangements/division for childcare/wraparound care?
HE hasn't mentioned any of that has he? I bet he thinks he's clever!
He's just giving you lip service hoping you will believe his shite.
I bet he won't make it easy for you to return to work and will play every game possible to manipulate you into believing that the 'hassle' is proof that going back to work is futile.
Then you'll be even more dependent on him.
If i were you i'd move the 'joint' money for your retraining into a separate account to "make sure WE don't accidentally spend it on something he deems urgent".
He doesn't seem to give a shit about your emotional needs or be very sympathetic to them.
How utterly patronizing and undermining is the attitude "fine, i'll help you get a job if it shuts you up"?
You NEED to prioritize your physical and mental wellbeing - and he is NOT being supportive.
So happily accept your dad's gift of money for surgery - your husbands attitude and help re dc/housework during your recovery from surgery will show you more of his real colours.
I would advise that you get counselling help too (not for the surgery!) to sift through and understand yourself better.
Your whole system has been impacted by the pregnancy and you're married to a man who is determined to bully and gaslight you.
I think he hates that you have the kind of parental/financial support that he's never had and he's bitter and resentful of it.
He doesn't want you to have anything of your 'own' and he wants to be in control of anything you do get.
He definitely needs to get counselling for his childhood issues and his bullying, controlling ways.
I might not want to sell my childhood home. I certainly won’t provide DH with cash to fritter away on luxuries
Sounds to me like you're realising what your husband is really about.
Re the house, i suggest your dad get a watertight, legally binding will/trust set up that protects his inheritance for the dc/you.
The only time an inheritance becomes part of the marital pot is if it's deposited into a JOINT/family account, and/or if you spend it on joint/family stuff like mortgage/renovations/holidays/bills etc.
So for now, any financial GIFT your dad gives you, make sure it goes into your personal account.
Then get legal advice for any future happenings.