  Epitaxial films of the B20-structure alloy Fe$_{1-y}$Co$_y$Ge were grown by
molecular beam epitaxy on Si (111) substrates. The magnetization varied
smoothly from the bulk-like values of one Bohr magneton per Fe atom for FeGe to
zero for non-magnetic CoGe. The chiral lattice structure leads to a
Dzyaloshinskii-Moriya interaction (DMI), and the films' helical magnetic ground
state was confirmed using polarized neutron reflectometry measurements. The
pitch of the spin helix, measured by this method, varies with Co content $y$
and diverges at $y \sim 0.45$. This indicates a zero-crossing of the DMI, which
we reproduced in calculations using first principle methods. We also measured
the longitudinal and Hall resistivity of our films as a function of magnetic
field, temperature, and Co content $y$. The Hall resistivity is expected to
contain contributions from the ordinary, anomalous, and topological Hall
effects. Both the anomalous and topological Hall resistivities show peaks
around $y \sim 0.5$. Our first principles calculations show a peak in the
topological Hall constant at this value of $y$, related to the strong
spin-polarisation predicted for intermediate values of $y$. Half-metallicity is
predicted for $y = 0.6$, consistent with the experimentally observed linear
magnetoresistance at this composition. Whilst it is possible to reconcile
theory with experiment for the various Hall effects for FeGe, the large
topological Hall resistivities for $y \sim 0.5$ are much larger then expected
when the very small emergent fields associated with the divergence in the DMI
are taken into account.
