This paper mainly focuses on cones whose basis is a maximum $h$-scattered linear set. We start by investigating the intersection sizes of such cones with the hyperplanes. Then we analyze two constructions of point sets with few intersection sizes with the hyperplanes. In particular, the second one extends the construction of translation KM-arcs in projective spaces, having as part at infinity a cone with basis a maximum $h$-scattered linear set. As an instance of the second construction we obtain cylinders with a hyperoval as basis, which we call hypercylinders, for which we are able to provide a stability result. The main motivation for these problems is related to the connections with both Hamming and rank distance codes. Indeed, we are able to construct codes with few weights and to provide a stability result for the codes associated with hypercylinders.