Dirichlet problem for what differential operator? By default I assume the Laplacian. With what kind of boundary data? By default I assume continuous.
Then smoothness is not needed for uniqueness: if two harmonic functions on a bounded open set have the same continuous extension to the boundary, then they are identical. The reason is indeed the maximum principle.
$C^{1,\alpha}$ is not needed for existence either. It would be odd to rule out rectangular boxes when considering boundary value problems. A sufficient condition for existence is the exterior cone condition: every point of $\partial D$ is the vertex of some (finite) circular cone which is disjoint from $D$. (This condition is not necessary, but it's good enough for practical purposes. The necessary and sufficient condition for existence is something you would not want to check in practice.)