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.