Is every relationship doomed to having the woman clean up after/nag her guy to a certain degree? And on some level, is this appreciated?

No, every relationship is not doomed to fall into this dynamic. it is important to remember that every individual person is unique to some degree. Values play into this in a big way; the acceptable standard of tidiness to which a room should be kept is one of those things that people don't think about consciously, and thus, the array of definitions that might fall into the category of "acceptable" often vary from one person to the next. This is one of the many areas where a relationship can benefit from clear and honest communication about values and expectations.