@archipelago: "finitely generated" refers to the totality of the integral homology groups.

Conjecture: nothing is known.

I think it is sometimes written $GSp_{2g}(\mathbb{Z})$, and called the group of symplectic similitudes.

@skupers: This is all for fields.

@YoungMath: As $\mathbb{R}^d$ is simply-connected, the local coefficient system $\mathcal{O}_{\mathbb{R}^d}$ must in fact be constant, so one can identify it (canonically) with the constant coefficient system given by its stalk at $0 \in \mathbb{R}^d$, which by definition is $H_d(\mathbb{R}^d, \mathbb{R}^d \setminus \{0\};\mathbb{Z})$. One then applies the usual UCT.