In differential cryptanalysis for DES, why is F(0...0) assumed to be always 0...0?
The question arises from a misunderstanding: The attack described in the paper does not work with actual inputs and outputs, but with differences between them. Hence Differential Cryptanalysis. 

Given two messages m1 = (l1, r1) and m2 = (l2, r2) with equal right halves r1 = r2, it obviously follows that r1 - r2 = 0...0 and F(r1) - F(r2) = 0...0.