In this work, we study the structure of linear, constacyclic and cyclic codes over the ring R = F-4[v]/(v(2) - v) and establish relations to codes over F-4 by defining a Gray map between R and F-4(2) where F-4 is the field with 4 elements. Constacyclic codes over R are shown to be principal ideals. Further, we study skew constacyclic codes over R. The structure of all skew constacyclic codes is completely determined. Furthermore, we introduce reversible codes which provide a rich source for DNA codes. We conclude the paper by obtaining some DNA codes over R that attain the Griesmer bound.