Determining structure factors for crystals from lattice and motif
Q1) A crystal is a convolution of a lattice and motif, so by the convolution theorem, the fourier transform of the crystal (i.e. your structure factor) is the product of the fourier transform of the lattice and the fourier transform of the motif. That is, this is general.
Q2) If the crystal is centrosymmetric* then the structure factor is real - this is a property of the fourier transform (the imaginary sine terms have opposite signs and equal other and are therefore zero). If your lattice and motif generate a centrosymmetric crystal then your fourier transform will be real. The motif doesn't need to be centrosymmetric to generate a centrosymmetric crystal.
*and the atomic form factors are real, but this isn't relevant.