Let me tell you what I know about the cohomology of congruence subgroups of Sp_{2g}(\Z). As far as cohomology with rational coefficients goes, this was determined by Borel. In the limit as g->\infty, it is isomorphic to a polynomial algebra with generators in degrees 4k+2. See his paper

A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. ´Ecole Norm. Sup.
(4) 7 (1974), 235–272 (1975).

I don't know of many integral calculations. I calculated H1 of the level L congruence subgroups for L odd and g at least 3 in my paper "The abelianization of the level L mapping class group", which is available on my webpage (click my name for a link). This was also determined independently by Perron (unpublished) and M. Sato. Sato's paper is "The abelianization of the level 2 mapping class group", and is available on the arXiv. He also works out H_1 for L even.

Another paper with information on H^2 is my paper "The Picard group of the moduli space of curves with level structures", which is also available on my webpage.

As a remark, both of the papers of myself mentioned above are really papers about the mapping class group and the moduli space of curves, but I ended up proving results about PPAV's and Sp_{2g}(\Z) along the way