  The aim of this work is to investigate the properties and classification of
an interesting class of $4$-dimensional $3$-Hom-Lie algebras with a nilpotent
twisting map $\alpha$ and eight structure constants as parameters. Derived
series and central descending series are studied for all algebras in this class
and are used to divide it into five non-isomorphic subclasses. The levels of
solvability and nilpotency of the $3$-Hom-Lie algebras in these five classes
are obtained. Building up on that, all algebras of this class are classified up
to Hom-algebra isomorphism. Necessary and sufficient conditions for
multiplicativity of general $(n+1)$-dimensional $n$-Hom-Lie algebras as well as
for algebras in the considered class are obtained in terms of the structure
constants and the twisting map. Furthermore, for some algebras in this class,
it has been determined whether the terms of the derived and central descending
series are weak subalgebras, Hom-subalgebras, weak ideals or Hom-ideals.
