Is a plywood backer required for all breaker panels?
Plywood is usually used when mounting a panel to masonry walls for a few reasons.
Isolation
Masonry is not a great insulator, which can lead to problems especially where the wall is below grade.  The cabinet has to be bonded to the grounding electrode system. Because of this, if you mounted the cabinet directly to a below grade masonry wall, you could end up with a ground loop.
Moisture protection
Masonry walls are also not great at stopping moisture. If the cabinet was mounted directly to the wall, there's a possibility of moisture in or on the cabinet, which can lead to problems.
Insulation
Because masonry is not a great insulator, a ground-fault could cause the wall to be energized.
Ease of Installation
It's much easier to attach electrical devices, cables, and raceways to plywood than masonry.

As for the color of the plywood, I've never heard of a requirement to paint it black.