The orientation of the dumb-bell which leads to the lowest energy possible for alpha>0 is along the z-axis. My reasoning behind this is that the net force working on one of the spheres will always try to bring the sphere as close to the center of the nearest ring as possible, meaning that the the configuration where the dumb-bell is aligned along the z-axis is a stable equilibrium. Whereas alignment along the plane perpendicular to the z-axis is also an equilibrium, it is not stable as a slight deviation would make it oscillate towards the z-axis.
For the case where alpha=0, every orientation of the dumb-bell has the same energy. As the quadrupole energy depends linearly on alpha.
For the case where alpha<0, the lowest energy corresponds to an orientation perpendicular to the z-axis. My reasoning behind this is similar to the first case, but now the distribution of mass of the rings is more skewed towards the side of the dumb-bell meaning that the net force will now try to bring the spheres to the side.