Did the peoples (Babylonians, Assyrians, Sumerians, etc.) of the Ancient Near East colour their wall reliefs?
Indeed they did. In fact, the walls of the main Assyrian palaces were covered with full color depictions of the various and brutal tortures the Assyrians would inflict on anyone that opposed them. It served as a grim reminder to foreign visitors and diplomats visiting the royal court as to the fate that would befall them if they dared to oppose the Assyrian King.
Yes they did. They had amazing skill with bricks, of all things, and the colors in some walls were fired right in. Check out the Ishtar Gate, for example. Ishtar Gate - Wikipedia
I'm not sure about all of those groups, but many of them, such as the Egyptians did.
Most of the time the paint is faded, chipped, worn or completely gone by the time archeologists rediscover something that would have been painted, whether through erotion or the sun