It is probably not OK to refuse to clean the kitchen. In exchange for a very valuable benefit, you're asked to do chores. Do the chores or pay for your own place.
Though, in my family, it's not like this. Members aren't valued based solely on what salable or cash contribution they make to the household. Everyone helps out proportionate to their ability, whether they bring in income or not - though bringing in income may mean a member has less time to help with chores.
But I'm not sure my family's attitude is healthier or better than yours. It has perhaps delayed independence, both financially and more broadly, for my little sister.
I think my family also wouldn't treat even a paid, non-family member this way - wouldn't make a task more difficult for someone else out of spite, or a sense of superiority, or "that's their job." My husband, however, used to never clear his tray at a fast food place; my family of origin always did. I married him anyway, and now he does. So, again, I don't know the moral value of making life easy for service people.
In my mind, it comes from a very anti-SF philosophy - that people are valuable beyond the monetary benefit they bring to me or to the family.
As a practical matter, it also has to do with coming from a family that had at least one young child for so long. You help out because that's what you DO in a family, no matter that your big sister can do it better or faster, or that the family managed fine before it had its youngest helper.
Think about it as your family seems to: If you were a paid housekeeper for non-family-members, and used your income to pay rent, and you felt your employers were being ridiculously messy, would you ask for more respect, or would you take your lumps?
Your situation just skips the middle step.
You want to live in their house ... then tell them how to use their own kitchen? And, if they don't use their kitchen to your liking, you'll flatly refuse to do your part?
No. You agreed to do the work. Do the work. If you don't like it, pay for your own kitchen and leave them to clean theirs.
Maybe. But, a normal adult person also pays rent.
Your example doesn't change my answer. Do your job, as agreed, or get your own place. If you're looking for permission to shirk your responsibilities, so far you're not finding it here.
Actually, in my house, they go in the sink, where they are rinsed with hot water, then a separate recycling bin.
It's somewhat unusual for a child of a family to have different standards of cleanliness and respect than the rest of the family, but everyone has their small bad habits and pet peeves.
Regardless of who's "right" in this situation, you have asked nicely for help, they've refused. Your options are to ask if you can stop doing chores and pay rent instead, or get your own place.
And a note on being "right" in adult family relationships: I've been married less a month, and I already know that being happy in your home life has very little to do with being "right."
A husband can be right or he can be happy but not both.
Do new brides get something in the mail or is it passed down by word of mouth?
It's in secret women-writing on the marriage license paperwork.