Thanks for getting back to this so fast.

A1: I calculated with the custom pupil function turned off and the result is nearly the same.

From what I understand about a Gaussian beam source is that each element of the pupil function is weighted by a ÔÇ£strength factorÔÇØ sqrt(cos(theta)) (├çapo─ƒlu, ─░. R., Taflove, A. & Backman, V. Computation of tightly-focused laser beams in the FDTD method. Opt. Express, OE 21, 87ÔÇô101 (2013)). This factor relates the field amplitude of the ray at the entrance pupil against the ray after the lens (facing the sample). I guess this explains the non-uniform patterns seen in the far field projection.
I think the strength factor is not considered in the far field projection because no lens is assumed in the far field projection, but only the rays coming out from the sample are calculated. I suppose that if I needed to calculate the field at the exit pupil of the collecting lens, I would need to divide by the strength factor again. Not 100% sure if this is right.
A2: Ok, I will use the rotation option, thanks for pointing this out.
A3: Ok.