Adding Level for non-repeated measurements

I have a cross-sectional (i.e., non-repeated measurements) dataset from
students ("stud_id") nested within many schools ("sch_id").
1- Given above, should we possibly add an additional random-effect for
"stud_id"? If yes, why?
2- Given above, should we also allow residuals in each school (e_ij) to
correlate? If yes, why? (I have a bit of a conceptual problem understanding
this part given the cross-sectional nature of our study.)

I think this is more a slightly-harder-than-elementary stats question rather than a "technical" query. If this was some types of 
GLMM, then the answer to 1 would be yes eg poisson GLMM then an individual-specific random effect adds in one type of 
extra-poisson variation. This is not the case for the gaussian (hopefully you see why). As to 2, consider how the *variance* of your 
measurement could be different within each school.