When I export object with material (screenshot 1) to .3ds or .fbx (2010-2012), it’s it’s imported by Unity 3D without texture (screenshot 2). Why?
Unity also don’t want to import .c4d file, it says
Cinema 4D couldn’t convert the .C4D
file to an fbx file. Try exporting the
.C4D file to an fbx file manually.
EXPORT_ERROR: FBX exporter not found
But I think, this bug is related to Cinema, not Unity, isn’t it?
Strange, i dont have any issues, the only real help i can give you is to reinstall either C4D or Unity.
Regarding Importing materials when you’ve exported a model, they need to be reassigned when you open Unity, To my knowledge you cant export textures from cinema 4D and into Unity, well, you can if you bake the material to a texture.
Have you tried exporting the model as an .obj wavefront file?, usually works like a charm for me 
To import cinema 4D files into unity, start unity as administrator to activate the importer.
You can put the c4d files with textures into the asset folder as well, so unity updates automatically.
FBX compatibilty between the different softwares is a common issue.
If you export FBX from Cinema 4D it will import correctly back into Cinema 4D but not for example in Blender or Unreal Engine. Also there is problem with smoothing normals etc.
But if you import the FBX into unity it works fine, if the textures are in the same Folder of the FBX file and or if they embed into the FBX file.
If you export from Blender it will import correct in Cinema 4D. So I assume it is a problem from Cinema4D.