We can see that the xx and yy component are the same, we know this means eta is zero. Checking the slide before this means the contributions of I+ and I- are zero.
The principal axis system is usually the one with the most symmetry. Because we take the origin in between the dumpbell and the axis such that one of them connects the two negative charges we have the most symmetry.
I think a toymodel with no axial symmetry could be for example one with a small charge added close, but not in the origin