You won't find a lot of info on the Fl. III's for the simple fact that they didn't work out well and the 100 vehicles that were supposed to be fitted out with the flamethrower units in 1942 were withdrawn from service after only 2 companies were outfitted with them for combat trials in Dec '42. The tanks were able to shoot flames only about 35 meters instead of the required 60 meters and they were converted back to normal gun tanks as a result or the chassis used for other purposes and weren't used in combat as flame tanks again.
As far as camo goes, the few available pics and the timeframe suggest either plain panzer gray or plain dunkelgelb (DY came into use in Feb '43 officially) depending. As for the carbon scoring, the flames were shot out of a 14.5mm tube at the end of the gun barrel so any carbon scoring would be minimal and only around the very tip of the barrel as a result. To replicate, use some black artist pastels or similar pigment and you'll be fine. HTH!