I am a SAHM to DC1 (5) and DTs (3). DH is out of the house 8-6 mon-fri.
DH gets up with the kids and get them started on breakfast as he is naturally awake early. I take over in time to clean up breakfast. I do all cooking, cleaning, washing, ironin, budgeting, shopping and any other house jobs. DH loads the dishwasher after dinner.
DH empties the bins, puts bins and recycling out, mows the lawn and helps 50:50 with the main veg gardening work (I do the weekday quick weeding and the actual harvesting for each meal).
DH does the kids baths, i get them into PJs and DH reads their bedtime stories.
DH will often cook once or twice over the weekend because he enjoys it.
DH will often take the kids out at the weekend if i need some down time.
if i ask him to he will do reading with DC1 (i only have to ask because he doesn't think to do it, he is more than happy to do it once i remind him).
when DTs were babies DH was much more involved with all household tasks without complaint. i did have to ask as he just doesn't notice the stuff that needs doing but he happily did it all when asked. once DTs were sleeping through i took on more and more again until back to 'normal'.
DH is very hands on with the DC when he is here - lots of rough and tumble here!
in terms of down time - DH is out one evening a week and every Saturday through the summer. I'm out one evening a week and get a few hours every sunday if i want/need them - he will either take the kids out or i will go out.
but like anything, it has to be what works for you - DH and i both like and enjoy our set up, it works really well and we both appreciate what the other does to keep the family in balance. you need to talk to your DH about how you feel and explain that while baby is little you need more support at home.