13. (1) The maximum number of points is 6. If $A$ is one of these points, let's assume $A$ is at the North Pole, then the remaining points must all be in the Southern Hemisphere (including the equator). If there is only one point $B$ at the South Pole, then the rest of the points are all on the equator, in which case there are at most $2+4=6$ points.

If there is no point at the South Pole, it can be proven that the number of points does not exceed 5. Otherwise, there would be at least 5 points $A_{1}$, $A_{2}, \cdots, A_{5}$ in the Southern Hemisphere (including the equator). Let $A^{\prime}$ be the South Pole, and the arcs $A^{\prime} A_{1}(i=1,2, \cdots, 5)$ intersect the equator at $A_{1}^{\prime}$. Then, among $\angle A_{1}^{\prime} A^{\prime} A_{\mathrm{j}}^{\prime}(1 \leqslant i \neq j \leqslant 5)$, at least one angle is less than or equal to $72^{\circ}$. Suppose $\angle A_{1}^{\prime} A^{\prime} A_{2}^{\prime} \leqslant 72^{\circ}$, then in the spherical triangle $\triangle A_{1}^{\prime} A^{\prime} A_{2}^{\prime}$, the distance between any two points is less than $\sqrt{2}$, which is a contradiction. Therefore, the maximum number of points on the sphere is 6. (2) The maximum number of points is 4. The proof is similar to (1).