I'm so sorry you are going through this. I do understand as I have been through something similar myself. I will try to answer some of your questions, but I am not an expert, except when it comes to dealing the the narcopath I married, but hopefully some of it will help.
He would often sulk for days (even with the kids) if he was unhappy about something
This silent treatment is typical of a sociopath, it is his way of punishing those around him. It is not your fault, but it is also not gaslighting. However it does come under the general umbrella of sociopathy.
If I didn't react the way he thought I should react to something, he would lecture me and treat me as though I was weird and wrong
Actually it's him that's weird and wrong. You are allowed to react in any way that is right for you. Yes, this is part of the gaslighters ethos - and it worked because it made you feel at fault even when you weren't.
He would often say things to friends and family designed specifically to embarrass me, even when I begged him not to. If I didn't find this funny he would say I had no sense of humour. Is this gaslighting?
Yes - again designed to make you doubt your self and continually question your own behaviour.
Breaks all of his promises
Again, typical sociopathic behaviour. Promises something which he has no intention of following through. He keeps dangling the carrot to keep you with him.
My younger kids at that point are having counselling as they're not coping. I plead with him, the counsellor pleads with him to wait. He waits a week and does it anyway
Sorry to say this but this is about him, not you or the children. Everything revolves around what he wants. He will only ever do things that suit his own agenda. Well done though for acknowledging that your children needed help. 
It's hard on them but with the right support they will come out the other end.
He denies or ignores any suggestion that he's not kept promises or behaved badly. Simply glosses over it or tells me off for insulting him
He's projecting his bad behaviour back onto you. You are a bad person for trying to make him accept his broken promises or bad behaviour. Mine told me I was being rude for calling him a liar. This was whilst he was looking at irrefutable proof that he had lied. Better to sling some mud back at us than deal with their own disgusting behaviour
Expects me to be grateful he's paying half the mortgage
In his head you should be grateful. He's pissed off that he's spending money on something not purely for his own benefit. I wouldn't thank him for meeting his obligations. As an aside, mine is so annoyed he's demanding that the mortgage payments for the last 10 years are returned to him. That really made me laugh.
Tells the kids I have loads of time on my hands and suggests I am lazy to them
This is part of the smear campaign. This is normal. The kids will make up their own minds on this one. I would not even give that thought head space to be honest (I would have a year ago, but I'm recovering from 30 years of this behaviour and slowly the light is starting to come through into my life).
He's already refused to allow her attend her art classes on the days she's with him as it's too much hassle
He is not child focussed. He appears to be self absorbed. This will not stand him in good stead for the future. This is not your problem, it is his. I do get that you're annoyed for your daughter, but again, she will soon make up her own mind about him if it becomes obvious that he will not let her do normal activities.#
He's beyond furious with me
Because you have stood up for yourself and your children. Well done.
Could someone please explain to me what this man is on about and how to understand what he's playing at?
This is the real crux of the matter. Stop trying to understand what he's playing at. You are wasting your valuable head space and whilst he's still in there, messing with your head, you are losing another little piece of yourself.
This is really easy to say (write) but I know that it's incredibly hard to do because I've been in the same position.
You cannot apply logic or reason to this type of behaviour. Your grey rock method will help to a degree, but that's what he sees on the outside. He's still in your head and you are thinking of all the things he's done and said to make you feel like you're the crazy one.
In effect you are shooting yourself with the same bullet over and over... And this needs to stop.
Acceptance is part of the healing process and once you accept that he's the crazy one, that will lighten your load.
Can you find some time and money to have some counselling yourself? It really helped me, although I've been weekly for a year now so it's been a long haul. However, my counsellor told me last week that a year ago I was 95% broken and now he thinks I am only 50% broken. It takes a long time to recover from this kind of emotional abuse but you can do it. Do what is best for you and your children.
If that means that you will no longer facilitate his manipulations, then that's the way it has to be. Your children do not need to learn that manipulation will get you want you want.
If I've written anything that comes across as harsh, I do apologise. But I really hope some of this helps a little. xx