First-principles study of bulk stacking, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>eff</mml:mi> </mml:msub> </mml:math> picture, magnetic Hamiltonian, <mml:math xmlns:mml="http://www.w3.org/1998/M

$α$-RuCl$_3$ is a candidate Kitaev material that exhibits zigzag antiferromagnetic ordering below 7 K. One contentious issue regarding this material is its bulk structure in the low-temperature phase. Recently, it has become generally accepted from experiments that the low- and high-temperature structures belong to the $R\bar{3}$ and $C2/m$ space groups, respectively. However, there was no theoretical study supporting the $R\bar{3}$-type structure as the low-temperature structure. In this study, we use constrained density functional theory to show that the $R\bar{3}$ structure is lower in energy than the $C2/m$ structure, in agreement with experimental observations. Then, we show that the conduction band minimum states are almost of the $J_\textrm{eff}=1/2$ and $m_\textrm{eff}=-1/2$ character, if we set the angular momentum quantization axis to be parallel to the Néel vector; this is the first analysis of the $J_\textrm{eff}$ picture for $α$-RuCl$_3$ from this perspective. In addition, we compute the anisotropic magnetic exchange parameters and $g$ factors of monolayer $α$-RuCl$_3$, thereby providing a comprehensive understanding of its magnetism. Our results demonstrate that both second-nearest-neighbor exchange interactions and magnetic moments not captured by the conventional atomic orbital projection method are necessary for accurate description of the magnetism in $α$-RuCl$_3$. Moreover, the calculated $g$ factors are in fairly good agreement with experimental measurements, especially the small anisotropy between their in-plane and out-of-plane components. Finally, we examine the effects of structural distortions from a perfect RuCl$_6$ octahedron, already present in bulk $α$-RuCl$_3$ without any external perturbation, on the magnetic properties. (The abstract is cut here due to the word limit; see the pdf file for the full abstract.)