We've got DSs (9&7) and the key to each is very different. DS1 needs one task at a time, and lots of reminders, and will carry it out 'in his own time' as his very perceptive preschool teacher mentioned. DS2 loves to be seen to be doing the right thing. He has just bounded to the door on my return from the supermarket to help carry the shopping to the kitchen and has unpacked and put away anything he could reach without being asked. DS1 is a great and loyal friend, a responsibility lots of adults can't handle. DS2 is a reliable team worker, a skill many of us could learn from. Not sure if that is learning responsibility and independence in the way you meant but it means a lot to us as a family.
As far as tasks go. They get up and get dressed for school, make their own breakfast and pack their school bags. DH makes their lunch for them to take (well, he does usually but he's broken his foot and is on crutches, so the DSs have added that to their morning routine this week - loads of praise for rising to the occasion and talk about the family as a team) and walks then to school (they've walked themselves this week for the above reason)
They don't yet take responsibility for instigating showers and washing (can anyone tell me if that happens with boys before adulthood) but do realise that teeth brushing is part of morning and evening.
DH is teaching them to cook (he's the SAHD and I do little cooking these days) by example. He's happy to let them help whenever he's cooking and they pick up techniques as well as getting Daddy's unadulterated attention in the process - great incentive
They are also encouraged to experiment in the kitchen which will stand them in good stead.
Homework is incentivised with time on a computer game they both enjoy. Incentivisaton seems to work and we're hoping that HW will become second nature. Wish it didn't have to just yet, but there you go...
I do wish some general tasks were a bit more engrained - not scattering dirty underpants as floor decoration etc - but I think the important things are coming along and some people are just never tidy so that might be hoping for too much. I'm guessing that by the time they leave home they might almost be human!
Good luck with your little monkeyDD. There's a lot to be said for not growing up too quickly. 