Deformable vs Rigid Remote Force can make a large difference because Deformable adds no stiffness to the structure while Rigid adds stiffness to the structure! In this example, deformable behavior allows the end faces of the rods to individually rotate while bending. Rigid behavior enforces the end faces of the rods to rotate as a rigidly connected group. I suggest you run the model both ways and look for the difference.
First question. A neutral axis is a concept used in the formulation of beam elements. A solid mesh does not have a neutral axis included in the formulation. Try to build the four rods from beam elements and see what you get. The forces in the solid rod elements make sense to me. Maybe when you flip the behavior of the remote force, you will get a different result that makes sense to you.
Second question. It is correct that the total force through a cross-section is independent of the mesh. I never said "the net nodal forces in a cross section depend upon how many nodes I have in a cross section". You asked "how can I find the maximum force (among all the nodes) at a specific cross section". I took that to mean you were asking about individual nodal forces and replied that you don't care about nodal forces because they depend on the mesh.
Bending moments apply to beam elements. There is no such thing for solid elements, which only know about forces. You can infer that the net resultant of the forces applied to the face of a solid mesh imply that a moment is being applied to that face.
Last paragraph. You design the area around a fixed support to avoid the stress created by Poisson's ratio by simply flaring out the material to a larger cross-sectional area. Below is an image of a rod flexure that illustrates this point.