Parameterized complexity of graph intersection number
The problem has been studied under the name Edge Clique Cover, and the observations you make regarding data reduction have been used to get a kernel with 2^k vertices. It is a long standing open problem whether a polynomial kernel exists. I don't know about good bounds on the running time, see http://theinf1.informatik.uni-jena.de/publications/clique-cover-jea07.pdf