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In $n$-dimensional hyperbolic space $\mathbf{H}^n$ $(n\ge2)$ there are$3$-types of spheres (balls): the sphere, horosphere and hypersphere. If$n=2,3$ we know an universal upper bound of the ball packing densities, whereeach ball volume is related to the volume of the correspondingDirichlet-Voronoi (D-V) cell. E.g. in $\mathbf{H}^3$ the densest horoballpacking is derived from the $\{3,3,6\}$ Coxeter tiling consisting of idealregular simplices $T_{reg}^\infty$ with dihedral angles $\frac{\pi}{3}$. Thedensity of this packing is $\delta_3^\infty\approx 0.85328$ and this provides avery rough upper bound for the ball packing densities as well. However, thereare no "essential" results regarding the "classical" ball packings withcongruent balls, and for ball coverings either.  The goal of this paper to find the extremal ball arrangements in$\mathbf{H}^3$ with "classical balls". We consider only periodic congruent ballarrangements (for simplicity) related to the generalized, so-called {\itcomplete Coxeter orthoschemes} and their extended groups. In Theorems 1.1-1.2we formulate also conjectures for the densest ball packing with density$0.77147\dots$ and the loosest ball covering with density $1.36893\dots$,respectively. Both are related with the extended Coxeter group $(5, 3, 5)$ andthe so-called hyperbolic football manifold (look at Fig.~3). These facts canhave important relations with fullerens in crystallography.

Top dense hyperbolic ball packings and coverings for complete Coxeter orthoscheme groups