It seems to me that, for $v = 6$, $e = 12$ and $f = 8$, the regular octahedron (dual-cube) is unique with such properties.
I am interested mainly about uniqueness of simple 4-regular planar connected graphs. If with these conditions and specified $v$, $e$, $f$ the uniqueness is not warranted, I don't want to figure out exactly how many such graphs there are.