  A family of spherical caps of the 2-dimensional unit sphere $\mathbb{S}^2$ is
called a totally separable packing in short, a TS-packing if any two spherical
caps can be separated by a great circle which is disjoint from the interior of
each spherical cap in the packing. The separable Tammes problem asks for the
largest density of given number of congruent spherical caps forming a
TS-packing in $\mathbb{S}^2$. We solve this problem up to $8$ spherical caps
and upper bound the density of any TS-packing of congruent spherical caps in
terms of their angular radius. Based on this, we show that the centered
separable kissing number of $3$-dimensional Euclidean balls is $8$.
Furthermore, we prove bounds for the maximum of the smallest inradius of the
cells of the tilings generated by $n>1$ great circles in $\mathbb{S}^2$. Next,
we prove dual bounds for TS-coverings of $\mathbb{S}^2$ by congruent spherical
caps. Here a covering of $\mathbb{S}^2$ by spherical caps is called a totally
separable covering in short, a TS-covering if there exists a tiling generated
by finitely many great circles of $\mathbb{S}^2$ such that the cells of the
tiling are covered by pairwise distinct spherical caps of the covering.
Finally, we extend some of our bounds on TS-coverings to spherical spaces of
dimension $>2$.
