Two topological spaces $ X $ and $ X' $ are locally homeomorphic if for any $ p \in X $ and $ p' \in X' $ there exists a homeomorphism from a neighborhood of $ p $ to a neighborhood of $ p' $.
Is every locally compact Hausdorff group locally homeomorphic to exactly one of the following?
i) Cantor set
ii) $ T^n $, where $ T $ is the circle and $ n $ is a cardinal
(I considered adding solenoid to this list but solenoids are not locally connected so I think they might be locally homeomorphic to the Cantor set. On the other hand, the Cantor set is locally totally disconnected which is stronger than just failing to be locally connected.)
A topological space is called homogeneous if it is locally homeomorphic to itself. Is every (locally compact Hausdorff) homogeneous space locally homeomorphic to exactly one of the above listed spaces?
No, this is horribly false. Indeed, a solenoid is a counterexample, since it is neither locally connected nor locally totally disconnected. For a simpler example of the same phenomenon, you could take a group like $\mathbb{R}\times G^\mathbb{N}$ where $G$ is a nontrivial finite group.
Another example is $G^I$ where $G$ is a nontrivial finite group and $I$ is uncountable; this is totally disconnected but not first-countable so it is not locally homeomorphic to the Cantor set. Or, you could take $G=S^3$ (or actually any compact connected Lie group that is not a torus would do) and consider $G^I$ for an infinite set $I$. This is not locally homeomorphic to $T^n$ for any $n$ since $T^n$ has a local neighborhood base of subsets whose third homotopy group vanishes but $G^I$ does not.
It seems extremely unlikely to me that there is any nice classification of the local homeomorphism types of locally compact groups in full generality.