Graphs whose maximal clique intersection graph has bounded chromatic number
For (2), any maximal clique $C$ of $G$ has size at most $\Delta+1$ and each vertex $v \in C$ participates in at most $2^{|N(v)|} \leq 2^\Delta$ maximal cliques. So the degree of $C$ in the clique intersection graph is at most $|C| \cdot 2^{|N(v)|} \leq (\Delta + 1) \cdot 2^\Delta$. Thus the clique intersection graph can be $(\Delta + 1) \cdot 2^\Delta+1$ colored. 

The bound can be tightened to  $(\Delta + 1) \cdot 3^{\Delta/3}+1$ by using the Moon Moser bound for the number of maximal cliques.