What a snidey, sneaky shite he is! He thinks he can get everything his own way, but hopefully, he's got another think coming. I'm really shocked that someone can do this when there's a newborn baby and you've all just moved house.
Definitely keep the door locked while you're home - after all, you don't feel safe on your own, do you? 
Or fit a chain, if it can't be locked so he can't open it. He's left, and the family home is now solely your residence. How disrespectful to think he can waltz in whenever he likes.
Get online and request a form for claiming tax credits. If you're on mat leave you're bound to be entitled to something. Here's the link [https://www.gov.uk/qualify-tax-credits]
Ask him how much maintenance he's going to give you for the children. And when you've copied those payslips, go onto the CMS calculator, and if they think it should be more, email him their figure and tell him this is the absolute minimum you will accept.
I'm indignant with rage about him expecting you to lie to your oldest. Just because he's a lying toad, he shouldn't expect you to start lying too.
You're behaving with great dignity (unlike my friend whose husband did something similar, he told Dfriend who the OW was after about 2 weeks, of course she was a colleague, so friend stormed into the office and shouted out "Get over here you fat minger" before dragging her out into the car park and giving her a good slapping!) and you should be proud of yourself for that.
And definitely don't express so he can take the baby out. That's an absurd request.
Stay strong, OP, you've had great advice on here. And he doesn't deserve you - you're clearly far too good for him!