Hello - I had *exactly* the same situation you describe with my ex-boyfriend (we just broke up a few weeks ago, and lived together for 2.5 years). Just as you said, he would do things around the house if I asked, but would never think to do it on a regular basis. The whole time we lived together, he never once cleaned the bathroom. I was the only one who would clean the bathroom, clean the cat litter, sweep/mop the floors, and usually the only one who did the dishes and the grocery shopping. The only thing he would do is the trash (we alternated weeks, and sometimes it seemed like I would do it more often than him), occasionally vaccuum the living room, and tidy up HIS stuff. He also (like your guy) thought of mowing the lawn as his job, and would always make a BIG deal about how much of a chore it was. And if the lawnmower broke or something, that would be his excuse to not do it. I just always thought it was really immature.
It's important to point out -- we both worked outside the home an equal amount (full time). So it's not like he was working and I was supposed to stay home and take care of the house. There was no reason why he shouldn't have been able to do as much as I did at the house.
Oh, another thing is, if I got mad at him about it (instead of asking him nicely to do something), he would get defensive and angry, and he'd say "If you're going to be like that about it, I'm not going to do it." I would get furious at that, because it shouldn't be like he's doing ME a favor by doing things around the house ... it's HIS job as much as mine. Grr!