Group cohomology of orthogonal groups with integer coefficient

For a precise answer to your first question, see Theorem 1.5 of

Brown, Edgar H., Jr. The cohomology of BSOn and BOn with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288.

For your second question, note that there is an isomorphism $PSU(n)\cong PU(n)$ for each $n$, and that the cohomology $H^\ast(BPU(3);\mathbb{F}_3)$ is worked out in

Kono, Akira; Mimura, Mamoru; Shimada, Nobuo Cohomology of classifying spaces of certain associative H-spaces. J. Math. Kyoto Univ. 15 (1975), no. 3, 607–617.

When $n$ is a prime, the additive structure of $H^*(BPU_n, \mathbb Z)$ has been computed independently in Kameko, Masaki; Yagita, Nobuaki, The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups ${\rm PU}(p)$ and exceptional Lie groups. Trans. Amer. Math. Soc. 360 (2008), no. 5, 2265–2284 and in Vistoli, Angelo, On the cohomology and the Chow ring of the classifying space of ${\rm PGL}_p$. J. Reine Angew. Math. 610 (2007), 181–227. For $n = 3$, the second paper contains a computation of the multiplicative structure.