The behavior for continuous charge distributions amounts to classical potential theory; for discrete charges, you get this behavior in the continuum limit.
It is true that the distribution is uniform for a sphere. On other manifolds things can be more complicated. See http://www.ams.org/notices/200410/fea-saff.pdf for a very nice exposition and further references. (In particular, Figure 5 shows some remarkable behavior on a torus.)
Torus Fig 5 http://cs.smith.edu/%7Eorourke/MathOverflow/Fig5TorusNearOpt.jpg
Incidentally, the behavior of 1 and 2 dimensions is not so strange. The charges do indeed end up on the boundary, if one uses a harmonic potential function (for example, a logarithmic potential in $\mathbb{R}^2$). The difficulty is that the Coulomb potential is not harmonic in $\mathbb{R}^1$ or $\mathbb{R}^2$. One way of thinking about it is that if you view the needle or disk as sitting inside $\mathbb{R}^3$, then the charges do all end up on the boundary, because the boundary in $\mathbb{R}^3$ is the entire set.
