Blondes
What a long journey you had, I'm so glad you have your DD now 
My DH is pretty good. I think it helped that he did most of the cooking before I got pregnant, and then started doing all of it when I had PGP and couldn't stand or walk for long at all without being in pain, and he's just carried on doing it. It also helped that he took a month off work after DS was born, and because I was breastfeeding, he did most of the nappy changes. Obviously that changed when he went back to work but he still does what he can. DH does bath time which is nice for them I think. He also does his share of rocking DS to sleep at weekends. But when I was ill and he looked after him for most of the day, he did have a shock!
I think what works is that we both have jobs that we "own" so we don't have to ask each other to do them. We both know DH will cook dinner so he just does it. I also think it's important for both partners to have roughly equal down time, and to work together to get the jobs done first before relaxing. It annoys me when DH is chilling out surrounded by mess because I can't relax until it's tidy (and I don't mean immaculate, I just mean toys chucked in the basket and dinner plates put in the dishwasher, that kind of thing)
purple
I think if your DP is really tired when he gets in from work it would be fair enough for him to have a short break, but it's not fair for him to sit on his arse all evening and leave you to carry on looking after the baby and house when you've already been doing it all day.
Do you get to go out once a week (an exercise class or seeing friends or something else)? If he has a weekly activity I think it's only fair that you have one too.
Don't be a martyr!!