Putting N points on a Spherical Surface

One could think of several ways to optimally put N points (call these "vertices") on a spherical surface:
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1. Such that the distance from any point on the spherical surface to the closest vertex to it (geodesic distance measured along the surface) is least ( this is basically the same as optimal covering).

2. such that the distance between any pair of vertices is maximized (same as packing).

3. such that the volume of the N vertex polyhedron formed by the vertices has max volume.

4. such that the polyhedron has the max surface area.

5. such that the polyhedron maximises the sum of edge lengths.

6. such that the arithmetic mean of the NC2 distances (measured along the surface) is maximum.

7. such that the geometric mean of the NC2 distances is max.

8. such that the harmonic mean of the NC2 distances is max.

Right now, let us stop with these 8.
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one can ask several questions now:
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1. For what lowest value of N are the arrangments achieving each optimum necessarily different (two arrangments of points are different even if N-1 of the points are identically placed but 1 is 'off')?

2. Is there any pair of questions above for which for all N, the same configuration of N vertices can be optimal? This looks unlikely.

3. Is there any pair (or triplet) of questions above for which for *infinitely many N*, the same arrangment of the N vertices can give optimal result?

4. Is there any pair of questions above for which one can have the same optimal arrangment of N values only for finitely many values of N? If so, are all pairs such?

And so on...
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One can also go from spherical surface to *any convex closed smooth surface* and ask analogous questions.