Well, you may find trends with oxide band gaps across the periodic table of elements which may be associated with the ionicity of the bonds, but there is no simple way to predict it exactly.

The most important reason for explaining the widening of the energy gap is the large difference in the radii of each of the cations and anions, and this leads to repulsion of energy levels, and thus the energy gap widens.

Am not sure whether the question makes much sense. The surface area depends on the preparation, i.e. for nanoparticles mainly on their size. The band-gap of ZnO is large compared to many elemental semiconductors but small compared to many oxides.