Painter's Servant

Each card becomes a new object as it changes zones, so this effect will apply to it from scratch in the new zone. However, replacement effects that care about the color of permanents entering the battlefield (like "[color] permanents enter tapped") will apply to permanents entering if that color was chosen for Painter's Servant because Painter's Servant's effect applies simultaneously with those permanents entering the battlefield. Zone-change triggered abilities that care about the new color (like "whenever a [color] permanent enters" or "whenever you cast a [color] spell") will also work.

The effects of multiple Painter's Servants are cumulative.

If something affected by Painter's Servant is normally colorless, it will simply be the new color. It won't be both the new color and colorless.