I'm still rather confused after trying a few more things.
I downloaded the table and chairs model from the warehouse either as SketchUp 2014 or 2015 version. Opened that in SketchUp and saved as version 8 model. Imported that in Rhino and in [my slightly modified] Rendered display mode it looks like this:
The shadows appear to be the result of the
... > Rendered > Lighting Scheme > Lighting method > Scene Lighting
setting that appears to be factory-default.
I first imported the 2015/2014 version model with "SketchUp Texture Writer" checked and placed one viewport in Rendered and one in Raytraced with this result:
Then, in another Rhino session, I imported the version 8 model and turned on the sun to obtain shadows (the Scene Lighting setting doesn't produce them here...).
I looks like all textures are read now but something's off with the scaling of the floor.
Then, when I go back to that other RH6 session (the one with the 2015/2014 version) and turn on the sun there, all of a sudden more textures are read by the Rendered display mode but not by the Raytraced mode. Turning off the sun to make textures go away again doesn't work.