I'm guessing you are trying to tackle a number of problems all at once. My suggestion is to focus on one simple thing and keep at it. I think it becomes overwhelming when we try to take on too many issues at once. I've experienced this many times, even recently, while trying to work out issues with one or both of my kids. I think it's far better to put a lot of effort into one thing and hopefully accomplish that goal than to try to attack several problems and make minimal progress.
Instead of expecting him to help when everyone is working on clean up, perhaps it would be easier just to get him to help you only and work out one or two things that you might do together. My son often is more helpful when I ask him to help me and we do the work together. If I ask to him to do something on his own, he just won't do it. Again, perhaps keep it small and build up little by little.
By the way, you might accomplish the dirty dish problem in a similar way to the dirty clothes problem. Designate the dishes that will be his. If he doesn't take care of them, he doesn't have the utensils or dishes to eat with. He might come around more quickly that way.
Hope there's something that you can use from my $0.05 or so.