This ability affects every card in every game zone, all tokens on the battlefield, and all spell copies on the stack, regardless of who controls or owns them.

This ability doesn't overwrite any previous colors. Rather, it adds another color.

The effects of multiple Painter's Servants are cumulative.

While Painter's Servant is on the battlefield, an effect that changes an object's colors will overwrite Painter's Servant's effect. For example, casting Cerulean Wisps on a creature will turn it blue, regardless of the color chosen for 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. Zone-change replacement abilities that care about the new color (like "[color] permanents enter tapped") won't work because those effects are applied as the card is entering its new zone. Zone-change triggered abilities that care about the new color (like "when a [color] permanent enters" or "when you cast a [color] spell") will work because those effects apply after the card is already in its new zone.

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.