It does. Paint tool it's on Edit Guides operator. This painting tool is used to paint stand channel that can be used to drive Ox operator parameters and soon for shading. So you will be able to use the painted valued on your shading network to paint the hair. You can find more information here and here. 

Ornatrix does not have a texture painting tool, only a strand painting tool. 

The plan is to implement strand channel data on all renderers but only Redshift is supported at the moment. 

No plans yet. As far as I know every host for Ornatrix have its own painting tool. C4D have body paint, rigth? And I guess you have access to an external painting tool.