  Kagome-lattice magnets $R$Mn$_6$Sn$_6$ recently emerged as a new platform to
exploit the interplay between magnetism and topological electronic states. Some
of the most exciting features of this family are the dramatic dependence of the
easy magnetization direction on the rare-earth specie and the kagome geometry
of the Mn planes that in principle can generate flat bands and Dirac points;
gapping of the Dirac points by spin-orbit coupling has been suggested recently
to be responsible for the observed anomalous Hall response in the member
TbMn$_6$Sn$_6$. In this paper, we address both issues with ab initio
calculations. We have discovered the significant role played by higher-order
crystal-field parameters and rare-earth magnetic anisotropy constants in these
systems. We demonstrate that the microscopic origin of rare-earth anisotropy
can also be quantified and understood at various levels: ab initio,
phenomenological, and analytical. In particular, using a simple and physically
transparent analytical model, we explain, with full quantitative agreement, the
evolution of anisotropy across the series. We analyze the topological
properties of Mn-dominated bands and demonstrate how they emerge from the
multiorbital planar kagome model. We further show that the most pronounced
quasi-2D dispersion are too far removed from the Fermi level, and therefore
cannot explain the observed quasi-2D anomalous Hall effect. By employing ab
initio many-body approaches, we demonstrate that the exchange-correlation
effects for itinerant Mn-$d$ electrons do not significantly alter the obtained
electronic and magnetic structure. Therefore, we conclude that, contrary to
previous claims, the most pronounced 2D kagome-derived topological band
features bear little relevance to transport in $R$Mn$_6$Sn$_6$, albeit they may
possibly be brought to focus by electron or hole doping.
