What and how much does coloring in the filament affect in the print? Are there general rules on how much a colored filament vs. vanilla filament affects the print results, what is affected and how those values should be changed in the slicer to achieve more similar results between different color/colored vs. pure filament spools?
At this point in time I don't think there is a need to be concerned with purity of filaments based on pigmentation.

For the most part, variations in filament quality due to coloring should be the least of your concerns compared to some of the other variables such as quality of the pellets, extrusion temperature (when manufactured), cooling rate (after extrusion), handling/storage, etc.

Also, assuming your focused on consumer 3D printer use, the typical hardware components aren't equipped to be accurate enough to make fine adjustments with regard to the quality range driven by filament color. Even if there were capable, accurate extruder(s) installed, I think you would need a well designed feedback loop to ensure that you're reading temperatures along the full extrusion process (drive, melt, extrude, etc).

I believe what you're asking involves more material science expertise, from a design aspect.

However, I believe that the more "color" you have obviously reduces the purity of the material and thusly the material properties can suffer. Such properties as thermal resistance found in PLA and ABS. So theoretically if you have Black filament, you'll want to extrude with a lower temperature than you would with a natural "White" filament. I would think that the necessary difference in temperature would be a few degrees (Celsius). However, there are many other factors, such as moisture and manufacturing techniques that can take precedence over color differences.

My advice, figure out how to "cheaply" analyze your material and ensure you have an accurate temperature feedback loop. If can you do that, you'll be able to drastically change the quality of consumer 3D printing.