As a paid personal care aide to my Grandmother, should I be expected to clean every moment she is asleep?

I don't think so. But tell her that if she wants you to tidy up the areas of the house outside of your grandma's room, you'll need more pay, because that changes the job description to housemaid, doesn't it? Since you have the time and could use the extra money, right? Just because you're a relative doesn't mean you can be taken advantage of; I stand strongly against that. :P