What is the relationship between minimum vertex cover and minimum clique cover
The minimum clique cover of the conflict graph is at most the minimum vertex cover of the original graph. In other words, if the original graph has a vertex cover of size $k$, then the conflict graph has a clique cover of size $k$. This is essentially because all edges adjacent to a vertex in the original graph constitute a clique in the conflict graph (though as the example of the triangle shows, there are also other cliques in the conflict graph). Details left to you.