Clearly any portion of a checkered board that can be covered by dominoes must have an equal number of squares of each color, but does the converse hold?

(Does identifiying opposite edges of the board to make a torus affect the answer?)

AND: Would someone please recall for me a problem that was posted here several months ago, but never answered, regarding properties of some kind of balanced two-color designs on a 4x4 square.

--Dan