@Janejones12 please tell us you’ve decided to move into the extension. It’s yours. You are in charge. You paid for it. You are a single mum to 4 kids so you are practically a living saint so you deserve it 
Your eldest son’s behaviour sounds worrying. You describe him as manipulative. You say he once threatened suicide. You describe how he smiles when he goes into the extension - ie more manipulation. You say he’s lazy and never helps. I’m afraid you need to do some more strict parenting with him. I suspect some of his behaviours have flown under your radar whilst you’ve been in survival mode. It’s time to help him become the good, kind, considerate, helpful, generous and genuine man he could be - but that needs a re-set of your current family dynamic - starting with you making it really clear who is in charge (you) and who gets the most privilege (you). If you give in to your eldest on this, you continue a pattern that rewards the behaviours you hate.
Your younger boy needs noise cancelling headphones and boundaries around x box use. If he’s talking or shouting too much whilst using it, he needs to know he only gets one reminder and then it goes off if he does it again. (Worked for my DS).
Your girls will very soon want and need their own space away from their mum. This is the opportunity to make this happen, take it.
Your sister is at best deluded, but sounds like she’s part of an abusive past you have escaped. You don’t have to please her and you shouldn’t try.
Your partner sounds like he has too much baggage. Ok as a friend but too much drama in tow for anything else. His opinion is interesting but irrelevant. It’s not his home, they are not his kids.
Ask your architect/builder to look at the two existing bedrooms and work out how to divide them. Then you decide which room is split for the boys. Tell them, don’t ask. Tell your girls which room they are getting.
Finally, lockdown has made many people reappraise their family lives and how they live. Don’t wait until everyone is back to normal. Take this opportunity before they go back to school to talk about the changes that you will be making permanently in your home and how everything will work come September. Timetable chores, who gets living room privilege/ screen time and when, so that it’s all fair. Make sure they all know that the extension is your space. It is not for gaming, playing, hanging out. It is yours. Enjoy it.