It is a well-known folk theorem that there is a homotopy equivalence between $(G\times EG)/G$, where $G$ acts by conjugation, and the free loop space of BG. My personal favorite proof is due to Kate Gruher, in the appendix to her thesis (Stanford, 2007). There's also a proof in the appendix to a recent paper of Klein, Schochet, and Smith (arXiv:0811.0771). The cohomology of the free loop space has been studied quite a bit in lots of cases. This KSS paper might have some useful information for you.

As Paul said, this fibration has a section, given by constant loops in the case of the free loop space model, or by sending $x\in BG$ to the pair $(1, \widetilde{x}) \in G\times EG$, where $\widetilde{x}$ is any point lying over $x$. This is well-defined since $1\in G$ is fixed under conjugation.

It is a well-known folk theorem that there is a homotopy equivalence between $(G\times EG)/G$, where $G$ acts by conjugation, and the free loop space of BG. My personal favorite proof is due to Kate Gruher, in the appendix to her 2007 Stanford thesis String topology of classifying spaces (ProQuest, subscription required). There's also a proof in the appendix to a recent paper Continuous trace $C^*$-algebras, gauge groups and rationalization of Klein, Schochet, and Smith (Journal of Topology and Analysis Vol. 01, No. 03 (2009) pp. 261–288, arXiv:0811.0771, doi:10.1142/S179352530900014X). The cohomology of the free loop space has been studied quite a bit in lots of cases. This KSS paper might have some useful information for you.

As Paul said, this fibration has a section, given by constant loops in the case of the free loop space model, or by sending $x\in BG$ to the pair $(1, \widetilde{x}) \in G\times EG$, where $\widetilde{x}$ is any point lying over $x$. This is well-defined since $1\in G$ is fixed under conjugation.