Where would one put $n$ telescopes on the surface of the earth
to see the whole sky as well as possible ?

Use the cosine metric to define how well we can see in direction $x$:

$ \qquad \text{cansee}( x; x_1 \dots x_n ) = \text{max}_i \ x \cdot x_i $

The worst direction for all the telescopes is then

$ \qquad \text{worstsee}( x_1 \dots x_n ) = \text{min}_x \ \text{cansee}( x )$

and we want $n$ telescope positions that maximize that,
i.e. that can see pretty well even in the worst direction.

That's in $R^3$. What I really want to do is generate approximate solutions
on a sphere in $R^d$ for $3 <= d <= 10$ and $d+1 <= n < 2d$.

"Very approximate" would do;
with < 20 points, an iterative method would do.

(Feel free to change the metric if min-max is intractable.)