Clique graph of bipartite graphs
I recommend you taking a look at this book chapter:


J. L. Szwarcfiter, A Survey on Clique Graphs, Recent Advances in Algorithms and Combinatorics,
CMS Books in Mathematics 2003, pp 109-136.


In there, many characterizations of clique graphs of specific classes are given. For example, it is mentioned that the clique graphs of chordal graphs are the dually chordal graphs and clique graphs of proper interval graphs are also proper interval graphs, and not trees and paths as you supposed.