Oh ffs. An adult should be capable of holding down a job and keeping their home clean and functioning.
Lots of us didn’t learn these skills growing up and had to figure it out. How did he manage before he had a partner to clean his toilet and wash his pants?
There are literally hundreds of women on here holding down jobs, keeping their houses functioning to a decent level and raising dc alone.
There are always going to be periods where one partner cannot pull half the load, or where the load gets heavier and people have to step up for a while.
I know it’s a MN trope to say “don’t have dc with this man” but seriously.
So yes, in short, he absolutely should be doing more.
I’m wading through the unending process of trying to secure meaningful help for my suicidal asd ds and I have such sympathy for what you must be going through trying to get help for yourself. Hugs 
But, rant over, LittleOwl makes a good point.
My list suggestion (for your Dh) is:
Everyday - do dishes, empty bins, pick up and put away, squirt bleach in toilet and do one load of laundry.
Once a week- change bed and wash, clean bathroom, hoover house, mop floors, clean kitchen counters (including under things) with an antibacterial spray.
That is achievable in under an hour.
I also suggest he writes out ten meals to use as a meal plan and shopping guide. Simplify lunches to sandwiches and soup. Simplify breakfast to toast and cereal.
That is the bones of household management in a crisis.