How to describe affiliation in between college and grad school?
Despite the differences in circumstances, I believe that the top answer in the linked question can be adapted  to your case as well.  Now, during the fellowship, and during graduate school, you will be registered with some university.  

The fellowship period is a little odd, but they had you register with a university for a reason, and it seems that you will be affiliated at least tangentially---unless people with your fellowship list themselves as an affiliate of the institution giving the fellowship, which happens in some cases as well.  Check and see what other people with your fellowship have done, and it should be clear.

During the gaps, use your last affiliation on publications, since you will almost certainly have done at least some of the work there.  As for describing your current situation during the gap, you don't have to say "unaffiliated", you can just be clear, e.g., "I finished at X in June, and am starting at Y in September."