Note: here $C_4(6)$ is the notation for the cyclic polytope of dimension 4 and of 6 vertices.
By the 2-neighbourly property of $C_4(6)$ and the Dehn-Sommerville equations, I've determined that the polytope has 6 vertices, 15 edges, 18 2-faces and 9 3-faces. Also, since cyclic polytopes are simplicial, their k-faces have exactly (k+1) vertices.
However, these are not enough to answer the questions above. What other characterization of cyclic polytopes that could help me with those questions?