Is the intersection of any two half-planes necessarily a half-plane?

In general no. The intersection of two parallel half-planes A and B is either a half-plane (either A or B, when A and B have similar orientation) or the empty set (when A and B have opposite orientation). When A and B are not parallel, their intersection is a maximal open region bounded by the two lines that define A and B, respectively. In this case, the intersection always exists and it is never a half-plane.