i dont know how important GRE scores are for foundation purposes.. but i think theres other way more important things in that application than GRE scores. they probably use them to see what are your "odds" of getting into grad school. If you have already been to, or are in grad school the GRE scores shouldn't really matter (unless you only wrote your name). Applicants for the fellowship are typically minorities right? some of us don't even have english as our first language, let alone answer those crazy GRE vocab questions.
and yes.. the department you're from should matter when they're looking at GRE scores, it does everywhere else so ford should be the same.. i dont see why they would expect the same scores for a physics student compared to a humanities student in math lets say.