I'm in a similar position - mobility issues, job that's not technically FT but often is, 2DC, and study.
I think your DH is already doing a fair bit, and I try not to complain if it's done in the 'wrong' way - look at the bigger picture, at least it's being done.
Your 7 year old can quite easily help out with sorting washing. Mine does. My 8 yr old walks up a flight of stairs carrying a washing basket with her and DS clothes and repatriates them, then brings the basket down. Perhaps providing separate small, stackable, baskets for each bedroom would help reduce weight and then you can sort them straight into the baskets from the dryer/line. In fact wee DS loves to empty the dryer and sort for me and he's only 4.
If you're already doing meal planning and shopping lists, then do an online shop, it's the next logical step. Takes about 20mins once it's set up. Stick the list on the fridge, and then when people feel up to it they can do a complicated/easy meal (I do a mixture every week).
Ironing - should only be DH shirts! Everything else (school uniforms, your own work clothes) should be either tumbedried and then folded, or hung on a hanger to dry. Voila, no ironing required. Don't buy linen and complicated fabrics like that. Consider whether you can afford to send the shirts out, usually between £1.20 and £2 per shirt.
He might be struggling too.