I have a question that is eluding an easy answer. Maybe somebody can help me. The question is quite straightforward.

Q) The viscous stress in a given direction is computed by the product of the stress tensor and the unit vector in the given direction. So, can the viscous stress normal to a wall be non-zero in the general case? Or must it only be zero, and the stress tensor take on appropriate values to satisfy this condition?

In the general case the normal viscous stress is non-zero. The importance of the normal viscous stress relative to other terms scales inversely with Re.