1) Because the matrix is diagonal, meaning the x- and y-divergence is the same and along the z-axes is another divergence. This is by definition ‘axial symmetry along the z-axis’.
2) The quadrupole term gives the “amount of change” along a certain axis. I don’t really see how the field would be equally diminishing in the xy-plane for all orientations of the dumb-bell. If oriented vertically along the z-axis, this would be trivial, but if the dumb-bell would be perpendicular (for example), I don’t see how the divergence has axial symmetry. Maybe by combining the fields of negatively and positively charges, but I don’t know how to derive this from the toy model (which does not have axial symmetry).