  We investigate the impact of mechanical strain, stacking order, and external
electric fields on the magnetic interactions of a two-dimensional (2D) van der
Waals (vdW) heterostructure in which a 2D ferromagnetic metallic Fe$_3$GeTe$_2$
monolayer is deposited on germanene. Three distinct computational approaches
based on \textit{ab initio} methods are used, and a careful comparison is
given: (i) The Green's function method, (ii) the generalized Bloch theorem, and
(iii) the supercell approach. First, the shell-resolved exchange constants are
calculated for the three Fe atoms within the unit cell of the freestanding
Fe$_3$GeTe$_2$ monolayer. We find that the results between methods (i) and (ii)
are in good qualitative agreement and also with previously reported values. An
electric field of ${\cal E}= \pm 0.5$~V/{\AA} applied perpendicular to the
Fe$_3$GeTe$_2$/germanene heterostructure leads to significant changes of the
exchange constants. We show that the Dzyaloshinskii-Moriya interaction (DMI) in
Fe$_3$GeTe$_2$/germanene is mainly dominated by the nearest neighbors,
resulting in a good quantitative agreement between methods (i) and (ii).
Furthermore, we demonstrate that the DMI is highly tunable by strain, stacking,
and electric field, leading to a large DMI comparable to that of
ferromagnetic/heavy metal (FM/HM) interfaces. The geometrical change and
hybridization effect explain the origin of the high tunability of the DMI at
the interface. The electric-field driven DMI obtained by method (iii) is in
qualitative agreement with the more accurate \textit{ab initio} method used in
approach (ii). However, the field-effect on the DMI is overestimated by method
(iii) by about 50\%. The magnetocrystalline anisotropy energy can also be
drastically changed by the application of compressive or tensile strain in the
Fe$_3$GeTe$_2$/germanene heterostructure.
