Just to reiterate that it's very important to take him off the electoral roll for your address - it's one of things creditors will look at when determining where he lives. You don't want that hassle coming to your door if he's applying for credit or not updating his insurance or driving licence details.
I would have steamed open that letter - it's coming to your address and I think you have a right to know what trouble he's potentially leading to your door. Then I would have returned to sender.
If he's driving around essentially uninsured because he's claiming to live somewhere other than his actual address, then he deserves to feel the consequences of that. Just passing on his mail still makes this your problem. He's thrown you under the bus numerous times, so I'm not against doing the same if the opportunity arises. Being continuously nice, just gets you walked all over.
I think I'd ask nursery if there's any way of them having the kids on the Fridays on your weeks only. If that's possible, it would give you that freedom to work.
I would also let slip to his mum that he owes you money. It doesn't hurt for her to see his behaviour as it truly is - money for tattoos but not for mattresses. When the time comes that you end up going for child support, it would be helpful if she is aware how this has come about, given that she's doing a big chunk of childcare for this lazy sod.