You work more than 12 hours a day.
She works part-time if at all (I'm guessing since you say unemployed or underemployed).
I don't think it's unfair to ask her to do more household tasks than you do.
My husband only works 8 hours a day and I do all of the household stuff except taking out the trash/recycling and some of the things that require a taller and/or stronger person (fixing certain things). He also takes care of the actual act of paying the bills, although I am the one who sets up the reminders in his email or leaves any paper bills we happen to get sitting on the desk with the due date plastered as large as possible on the envelope..
I feel like me doing the laundry, dishes, picking up the house, vacuuming, dusting, etc. is a fair trade for him working full time and pretty much fully supporting me. Because of his willingness to do that, when he's off, we get to spend time together OR he gets to goof off and have fun. That's the way it works. We both work about 40 hours a week, mine just happens to be in the home - either keeping up the house or doing random odd jobs for a bit of cash.
Seriously, nothing you have said sounds irrational to me. Granted, that's coming from another "high-maintenance" person, but you're not imposing on her life at all by asking her to do a few things. You're just trying to maintain a partnership where both people contribute TO THE HOUSEHOLD and it doesn't sound like she wants to do that. Sure, she may be dedicated to your romantic relationship by supporting you emotionally and caring about you as a person, but is she dedicated to the relationship you two have as cohabitants?