Sorry but I think you're being naive and overly optimistic.
Get a lawyer ASAP and a good one at that (ask friends & family who's divorces went fairly well for recommendations or even name change & ask on here with location)
"Can I start the process to claim additional benefits while he is still in the house." In theory you can but all finances & living arrangements have to be as they would be for 2 flat mates sharing not in a relationship and from your description I can see him telling dwp you're lying.
"Am I entitled to stay in the house if I take over the mortgage payments?" Not necessarily - you REALLY need legal advice on this.
"Is it reasonable to expect him to help finantially with the children as I'm the one ending it? And does this include nursery costs?" Its reasonable to expect it, it's highly unlikely to happen. Is he employed? Self employed? Even getting maintenance can be difficult.
He is NOT your ally any longer, don't tell him anything you don't have to.
1 open a separate bank account of your own with a completely different banking group to his
2 get a good lawyer. Do NOT agree to anything without checking with lawyer first. NOT using a lawyer could well end up far more expensive than using one. Especially with a house involved. It's utter folly to divorce without using a lawyer if there's children, property involved. Any decent lawyer will tell you that. You don't have to use one you're not comfortable with though.
3 gather all important papers and keep safely (wedding and birth certificates, passports, bank & savings statements, mortgage papers etc)
4
"You don't need to prove it" sorry but that's rubbish! Otherwise married & co-habiting couples all over the country could claim fraudulently. I'm a Lp and I've regularly had home visits to check I really am living alone even with no reports made. They are entitled to check bank statements etc too.
You may well feel emotionally stronger once you know where you stand legally & financially than if you try to talk to him now.
Anyone who says divorce is easy is lying or hasn't been through one. It can be straightforward but it's never easy, nobody marries thinking they'll divorce.
But that doesn't mean it isn't worth doing. Mine was 2 years of hell but i don't regret it at all
"he's been kicking and punching doors, screaming at me" that's violence. You can call the police when he's doing this. Also if you get a record of dv establish you might be able to get legal aid for the divorce.
These people are apparently good in situations like yours. Apparently can be particularly helpful in getting abusers removed from the marital home. I believe it's a service run by lawyers.
www.ncdv.org.uk
"I also got on the laptop to look for the spreadsheets on house running costs / mortgage etc but they have gone. He must have deleted them. There is a word document from our mortgage provider but it's password protected." Nothing is ever really deleted. Sometimes if you email from a computer with deleted files on when you go to do an attachment the deleted or old copies of those files can be available to add as attachments (I've done this when I've accidentally deleted stuff) other more techy mners might be able to tell you other ways to track down deleted documents.