Is the cohomology of the genus 2 mapping class group (that is, the cohomology of the moduli stack $M_2$ of genus 2 curves) known? I'd be interested in references. The rational cohomology is known to be trivial, but I am interested in torsion phenomena, especially at the prime $2$.

The possible automorphism groups of a curve of genus two are known very classically, this is usually attributed to Bolza. In his list we find only three prime factors in the orders of the groups: 2,3 and 5. So this is the only possible torsion. I haven't read Benson and Cohen, "Mapping class groups of low genus and their cohomology", but the Mathematical Review suggests that it is exactly what you are looking for. They compute the ring structure on the mod 3 and mod 5 cohomology and the Poincaré series of the mod 2 cohomology. 

