We compute the Poincaré polynomial and the cohomology algebra with rational coefficients of the manifold $M_n$ of real points of the moduli space of algebraic curves of genus $0$ with $n$ labeled points. This cohomology is a quadratic algebra, and we conjecture that it is Koszul. We also compute the $2$-local torsion in the cohomology of $M_n$. As was shown by the fourth author, the cohomology of $M_n$ does not have odd torsion, so that the above determines the additive structure of the integral homology and cohomology. Further, we prove that the rational homology operad of $M_n$ is the operad of 2-Gerstenhaber algebras, which is closely related to the Hanlon-Wachs operad of $2$-Lie algebras (generated by a ternary bracket). Finally, using Drinfeld’s theory of quantization of coboundary Lie quasibialgebras, we show that a large series of representations of the quadratic dual Lie algebra $L_n$ of $H^*(M_n,Q)$ (associated to such quasibialgebras) factors through the the natural projection of $L_n$ to the associated graded Lie algebra of the prounipotent completion of the fundamental group of $M_n$. This leads us to conjecture that the said projection is an isomorphism, which would imply a formula for lower central series ranks of the fundamental group. On the other hand, we show that the spaces $M_n$ are not formal starting from $n=6$.