DH and I are both avid gamers, but we prefer quite different types of games, so we don't play together. If I can, I like to play for a few hours at a time on weeknights (unless I'm too busy from work or whatever). We've found it works best, though, if we play at the same time (him on his computer and me on mine) so that neither of us feels neglected while the other is engrossed in something. Scheduling gaming time like Allyson suggests could also work well for you and your potential future partner. It can easily work well with a non-gamer, just like any other big hobby, as long as both parties make the effort to make it work.
With dishes and so on, scheduling has worked very well for us. I'll always wash dishes immediately after a meal, before I do anything else. In the rare cases when something has to be left to soak, it gets washed after the next meal, no later. Right after brushing my teeth each morning I allocate 5 minutes to cleaning--usually involving me running around like a madman with a dust mop. DH has similar things that are his jobs to do. For both of us, as long as we stick to our schedule, household chores get done without much effort and we get time to spend with each other. I feel it isn't another person's job to remind you what to do once you're an adult, and you should work on organising yourself so that at least most of the time they don't have to be burdened with this unfairly.
Sundays are our "go out and do something" day. We have a list of local things that we haven't seen/done (compiled largely from Trip Advisor, although we add to it whenever we see an ad for something interesting). Come rain or shine, we'll go and do something on that list every Sunday (various specialist museums, parks, zoos, etc.) which ensures that even if we've both had a busy week, we'll get to spend some time relaxing with each other while experiencing something new.