4dc here too but older (13-5) and dh works away early Mon-Thurs night.
Shopping is done online. I have a pad in the kitchen for writing a list as I go (everyone adds to it) and probably average 2 orders a week.
Non-iron school shirts so the only ironing is 4 of dh's shirts and any of my stuff I want to do.
I have 3 trugs on the landing for dirty washing, when one is full, I wash it on a 40 auto wash - I rarely sort lights/darks as that seems to be when I get a back log. Then straight into drier then sorted I to piles on my bed. Everyone puts their own clothes away.
The big 2 are expected to do one stage of the laundry per day. Everyone us expected to put their dirty washing in the trug.
All 4 dc can empty/stack the dishwasher and all help with the bins. They each tidy their rooms on a Sunday night - until recently we had a cleaner on a Monday (now it's back to me) - so that they can be hoovered without needing tidying first.
Lots of easy storage helps to - big box for Lego, big box for 'kitchen', big box for wooden train and so on so that it's easy for the dc to tidy up themselves. With the younger 2, they cannot just tidy their room, I need to tell them to put the lego away, then put the Barbies away etc - break it down into smaller jobs.
If the mood takes you, choose a room and hit it - I'm not a hoarder so can be ruthless about clutter/stuff. I probably take a small carrier bag a fortnight to the charity shop of bits that we just don't need (outgrown clothes, books that we've had enough of, toys that are ignored). Don't keep every single bit of stuff the dc have made - choose the best things and display them (pictures blu tacked to kitchen cupboards) until something 'better' comes along.
That's really long, sorry 