Good questions; I think your impressions between Fulbright and DAAD are spot on (or at least they confirm what I had thought).  In regards to the university affiliation.  I don't know exactly, but I am doing straight research to complete my degree here in the states.  I will be working with a professor at FU Berlin, but not enrolling in a path for a degree there. I stated that I would travel to different archives throughout Germany, making brief research trips, but I don't know about working at multiple institutions that might be a different situation.  Hope that helps or gives context at least!