  Cubic perovskite-structure ABO$_3$ and A$_{1-x}$A$^{\prime}$$_x$BO$_3$-type
oxides have been investigated extensively while their hexagonal-structure
versions have received minimal attention, even though they are multiferroic and
can form heterostructures with the manifold hexagonal two-dimensional
materials. Hexagonal ferrites of the form RFeO$_3$, where R is yttrium or a
rare-earth element such as Lu, Yb, etc., feature coupled ferroelectricity (FE)
and weak-ferromagnetism (wFM), exhibiting linear magnetoelectricity. Their only
drawback is weak ferromagnetism. In this paper, we employ
density-functional-theory (DFT) calculations on hexagonal LuFeO$_3$ ($h$-LFO),
targeting its magnetic ordering by electron doping,anticipating
spin-disproportionation of the Fe sublattices. Indeed, we show that
spin-disproportionation in heavily-electron-doped versions
Lu$_{1-x}$Hf$_x$FeO$_3$ ($h$-LHFO), especially for x=1/3 and 1/2, leads to
robust out-of-plane collinear ferrimagnetism that is stable at room
temperature. Furthermore, the robust ferroelectricity of $h$-LFO persists via a
Jahn-Teller metal-to-insulator transition. Finally, we construct a
$h$-LHFO/$h$-2D heterostructure, where $h$-2D stands for the FE/FM monolayer
MnSTe, and demonstrate strong magnetoelectric coupling, namely manipulation of
magnetic skyrmions in MnSTe by an external electric field through the $h$-LHFO
polarization, opening up a new realm for magnetoelectric applications.
