Does there exist a topological group where every point has a (nontrivial) neighborhood which is a subgroup? Does there exist a topological group which can be covered by (nontrivial and proper) open subgroups of itself? If so, what are groups of these types called and is this a nice property for a topological group to have? Or is this just impossible?

It's certainly possible, consider $G = \mathbb{Z}^2$ with the discrete topology. If $x \in G$ is any element, consider $H$ to be the subgroup generated by $x$ (unless $x = 0$ in which case consider $H = \mathbb{Z} \times \{0\}$ for example). It's proper because $G$ has rank $2 > 1$, nontrivial, and open because $G$ is discrete.
If you want an example that's not discrete, consider $G = \mathbb{R} \times \mathbb{Z}^2$ and apply a similar reasoning. I don't know if there's a connected example and I don't know if this property has a name.
