Not read the full thread - got to page 4 so sorry if repeating anything already said.
Yes it’s expensive when you have very little money. If I where you I would scrape together the up front rent for the council property and hand in notice for your current place ASAP.
Washing machine - I would advise using the good will of family and try to find a laundrette. Hand washing small things like underwear and socks. That should buy you a few weeks / month or so to save up for something - even if it’s second hand.
Cooker - see if anyone you know has a camping stove or a table top job you can borrow for a while. That combined with a microwave (if you have one) might see you through.
Fridge - more problematic so I would prioritise getting this first.
Carpets - do you have rugs? You can put them down to make do. Most pound shops also have the sticky floor tiles. These would do for smaller rooms and don’t work out too expensive. I floored my kitchen for £12 and my bathroom for £8 (just do the main area initially and leave under appliances.
Sofas- any seating will do to start with. Check gumtree for free or very cheap. Ask around friends and family, you might not get something pretty but a couple of cheap throws can cover and ugly / busted sofa until you can afford to replace.
You will also need things like curtains which you can get in charity shops for cheap.
My best advice - check gumtree for freebies or very cheap. Scour charity shops. Check local Facebook selling pages and even post asking for the things you need (many people do give away things they don’t need if they know it’s going to someone who will appreciate it).
There is the other route of buying stuff and paying it up (catalogues and shops like brighthouse) but they are the expensive option as they charge a fortune in interest. I know a lot of people on low incomes that do this as you only pay a small weekly amount over something like 5 years. It is an option if you are really stuck but not one I like to advise much as you do pay over the odds in the long run.
I hope it all works out for you OP.