I am a little bit behind when it comes to color and texture painting in Curvy. Is it possible to actually edit/paint on such a texture channel straight within the Curvy environment, in order to "paint in" small extruded 3D details on the surface of a model ?
Yes - absolutely. Originally painting on a live "Curvy Map" was the main way to sculpt a model in Curvy.
It's a good idea to load a grid onto your model first to check the UV maps are setup OK. Then load a blank image or Flood Fill the map. Use the first Paint Brush in the Tools Panel and you can paint directly in 3D or in the 2D viewports.
If you click on the Curvy Map you can paint on that too - just use black and white as paint colours. Smudge brush (for texture) works as well.
I might use this more as a sculpting tool again - as it lets you easily "undo/fade" a change to the model by painting in 50% grey.