DH is very hands on with the kids, lots of playing with them, swimming, bike riding at weekends.
On a daily basis he empties the dish washer and clears the draining board in the morning before he leaves for work (at 6.30am for a 12 hour day).
Evenings he listens to home work reading, does Pjs and bed time, bath if it's bathnight. I am often out the door to gym or college as he comes in. He will have a quick tidy if I remind him.
He also does all the ironing once a week. Puts the bins and recyling out, empties indoor bins.
Other than that he will occasionally hoover or clean bathrooms, especially if we have visitors. He has done a big tidy and sort out today, as he was in the mood.
I occasionally have an explosion about being the only one who ever picks anything up in this house (DH been know to "step over" the kids mess) and he is getting better about tidying up as he goes.
I have 2 part time jobs, both working from home, 1 preschooler, 1 school kid plus part time care of my nephew.
I think we have a reasonably good balance.
Hope your DH gets a job soon and you find a balance with tasks around the house.
Just a thought - DH read a report saying that men who help around the house get more sex (woman feeling more like sex as she is not so fed up and disgruntled at doing all housework). He has been more keen to help out since...