I think TheInvestigator is right.
Unless you have it writing that you 'd pay mum back, or she part owns your house now, just say "well go then, if that's what you want" next time she threatens, as that's not healthy. And No she doesn't get £80k back, it's yours & your children's house, not hers: "we'll call that rent & care for 4 years"". It would stand up in court as she gifted OP's brother similar amount and then lived with op at a time she needed support. ( A care home would have been £600-800/week).
Then you'll be on equal footing and she'll be a bit more respectful to you in your home. If she's not washing or cleaning up after herself in shared kitchen or bathroom knock on her door and say , "can you come out here and clean up your mess?", same as she would have done to you when you lived in her house.
And it's ok to say "Stop ✋, don't criticise" & walk away, if she's ranting on about a parenting decision you've made.
I'd ignore as much of annoying stuff as possible , the fox thing would have made me lol!! Or turn it back on her, "mum are you sure you want to go out in that flimsy coat in this weather? ... I'm not sure about your friends... Have you been to the toilet?" Give her back a list of patronising questions each time she criticises and she'll soon learn!
Really you hold all the cards here. It's lovely mum is living with you, that she's getting out and about living her life after being widowed. And that you've supported her through a tough time in her life, but you don't have to silently continue to put up with lots of selfish or critical behaviour. Call her out on it, same as you would a teenager. She's only 60, she's not old!! She can get herself to GP appointments if need be.
Work out what is fair share of bills. She's living far cheaper off you than in her own place. So say, right we can't keep subsidizing you mum, your share is £250/ month and please don't touch the thermostat /put the heating off in day or you'll need to pay most of the gas bill, as we can't afford how much it's jumped up to. (Tbh why don't you adjust timer so it stays in a bit longer and comes back on earlier, to compromise)
Overall you sound a bit disappointed that she's not become a doting grandma or a more loving mother. But she's the mother you've got. Try to find what you do like about her or it'll be a long old 20-30 years of her living with you. Ps. You can put her name down on the Local authority/borough (council) housing waiting list for a rented sheltered flat 😁