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Yes to both questions, assuming that the group $G$ is locally compact and Hausdorff. In such a group one can always find an open subgroup $H$ which is isomorphic to $(K\times L)/\Gamma$, where $K$ is compact, $L$ is a $1$-connected Lie group and $\Gamma$ is a discrete subgroup of $K\times L$. This reduces both questions (or assumptions) to the compact group $K$. No topological countability assumptions are needed.