Is there any known result on the maximum degree of faces in regularandplanar graphs ? In particular, is anything known about maximum degree of the faces in a 4regular planar graph? By degree of a face I mean the number of edges forming it.

This will likely only serve to sharpen your question, but I will just observe that there
is no upper bound on the number of edges of a face of a 4regular planar graph:



It makes a huge difference what restrictions are imposed. Without any restriction on connectivity or edge type, you can put every edge on a single face (a path of double edges with a loop at each end). With 2connectivity imposed, you can get half the edges (a cycle of double edges). If you want 3connected graphs, the antiprism (shown in Joseph's picture) gets you half the vertices and I think that is optimal when the number of vertices is even. 

