2 more cents: 'I don't do X' isn't the same as 'I don't like X.' The verb 'do' does imply action, so a refusal, not just a disinclination. Normally even if you don't like mornings you still manage to get up, make small talk, get dressed, go to work, etc. But if you're the legendary Garfield, you simply decline, refuse to function, just say no. Cats, slugs and grouches make their own rules.
Some typical contexts:
'I don't do windows/dishes.' Probably the original. At first simply defining real duties, since the better class of servant didn't do messy scullery jobs. Even today, not just an inclination but an actual refusal: I may be little more than a lowly wage slave, but some crummy jobs are still beneath even me. However, used ironically, 'I don't do photocopying' or 'I don't do toilets' could also just be an empty protest (like 'I don't do Mondays.')
'I don't do drugs.' Another habitual action, not just a preference. The idiom with this verb is only for harder drugs: Do heroin, do cocaine, but drink alcohol, smoke pot.
'I don't do subtle.' Here another layer of original context might also be acting, casting: Can you do sad for me? She does menacing really well. He can't really do dignified, he's too oriented toward comedy.
That last may overlap with the rather British 'offer' context, as in hotels, restaurants, pubs: Their breakfasts aren't up to much, but they do lovely high teas. It might imply something a little fancier than usual, a finished production or presentation, a special bit of repertoire (great word, perfect tie-in with acting).
In the US, the closest similar context might be cooking: Can you do an egg-white omelet? He does a fabulous lasagna. But that's basically just 'make' in another guise.
Could the common denominator of meaning be doing something harder than average?