Question

A metal rod moves at a constant velocity in a direction perpendicular to its length

Answer 1

The velocity is not parallel to the field and the rod is parallel to neither. The electrons and protons experience a Lorentz force which produces an emf in the rod.

There is some illogic in the usual description of the Lorentz force. The velocity is usually described as being relative to the field lines.

But field lines are only a graphical metaphor, and cannot constitute a reference frame, especially in a uniform field. In practice the velocity is relative to the atoms that produce the field.

This suggests to me that the assumption that the Lorentz force can occur in a truly uniform field might be an unwarranted extrapolation.