My GPS is a hand held battery operated unit so it should work. An EMP attack would fry it though. I don't know if the GPS system relies in any way on the electrical grid but I doubt it since its main purpose is for military use. Again, an EMP attack could or would fry the whole system not just the user's personal unit. That is why I still carry an old fashioned compass.
(Dumb question probably.)
I don't think so. ;-)