How old are your children, OP? You don't say, but from the sound of things, your DD is only very little still.
So, 1st things 1st, sort out sleeping. Take your three 'spare' rooms - the two bedrooms and the living room.
SS goes into smallest room on his own. Decorate it to his taste, and it becomes his, with his name on the door. Yes, he's potentially a bit old for that, but you're working to give a sense of permanence and belonging in the home.
Middle room becomes 'family' space, with sofa, tv, etc.
Largest room gets spilt into 2. Have a look at options 4 and 5 here to do it. One side is yours and partners; the other belongs to your DD's.
You'll share with baby for now, but set it up to move her across as soon as she's ready, and store her stuff in there already. If you go for the cube store unit that will help here as they store HUGE amounts of stuff.
Bedroom's become out of bounds for anyone but their owners, so your SS is NEVER allowed in DD's 'room' and your DD's are NEVER allowed in SS's room. It's up to you whether you enforce the same rule with your 'room', but if you do, it has to be for SS and DD equally.
That sorts your living space out for the moment.
Keep going with the home based tasks - yes, he'll do them badly. He's an parented 11 year old boy, but it's not the result that matters. It's him taking part in running your home and family, and having a routine and small responsibilities.
Is he year 6 or year 7? Either way, you need a meeting with his secondary school as soon as term starts. Explain the situation, and insist on them starting to look for support.
Take him to the GP and ask for counselling there as well. He needs to talk. His mum bailing on him will have been traumatic.
Get him into at least one club that meets every week, preferably something with structure that he's good at.
You may not be seeing it, but you are already making a difference - you said his school work and behaviour have improved already. That's a very positive sign.
It's going to be a long road, with backwards as well as forward steps, but be kind and fair and above all, consistent, and you could make all the difference to this boy's life.
All that said, however, you need to be prepared to move out with your two DD's. There are some red flag behaviours here, and if you can't resolve that living together - e.g. he keeps intruding on your DD's space, pulling covers off her etc, then you need to act to protect all the children.