Well from your update he does sound quite shit.
You need to set up a small desk in your bedroom for WFH - I still think working in the living room is not fair.
He sounds like he’s skiving off looking after your child - 45 minute poo break, fag break. You are making yourself more available by being in the shared living room, and by humouring him.
Go into bedroom at 9am to work, tell him only to disturb you if it’s really necessary, then close the door. When WFH, boundaries is important. That includes not asking him to go fetch you food or interrupting him/winding up LO when he is doing childcare btw, that sounds really selfish and immature. If you were really that ill you shouldn’t have been working or looking after LO at all.
He does sound very lazy and useless, but you need to stop being a martyr and let him fail. Little one isn’t going to starve to death from missing lunch. He’ll realise his mistake when she starts kicking off, and have to detour to the shop for snacks. Just let him get on with it.
The car situation sounds bizarre. If his car isn’t safe, then yes you need to let him use your car to take your child out in the day (both are family cars, surely?). It’s not fair for him to be stuck in the house all day with little one (with you working in the living room!!!) just because you want to use the only good car at 4pm. It does take a long time to get a child ready and out of the house sometimes. Having the 4pm deadline hanging over him is really annoying, I can see why he’d think you were being unreasonable and it made it pointless for him to take the child out at all.
You need to sort out your WFH routine. But it depends - who normally does childcare when you’re both working? How frequently does he do childcare while you’re WFH? More context required I think.