Whenever I create a new 3D extrusion the extruded part or the width is always gray. Que 01 > Can I do something so that my image wraps the whole thing? Que 02 > Can I give it custom color?
To give custom colour you have to select the new extruded object and in the properties panel there is defuse option. Changing that colour will change the texture as well.
If you want to add texture, you can click on the small folder Load or Generate a new one. By generating one, you will have sort-of 'smart object', to which you can add your images and options. You can add few more textures for Specular, Illumination, Normal etc. by click on the folders next to the sliders.