  We analyze the impact of the $\theta$-angle and axion dynamics for two-color
(in fact any $Sp(2N)$) QCD at nonzero baryon charge and as a function of the
number of matter fields on the vacuum properties, the pattern of chiral
symmetry breaking as well as the spectrum of the theory. We show that the
vacuum acquires a rich structure when the underlying $CP$ violating topological
operator is added to the theory. We discover novel phases and analyse the order
of their transitions characterizing the dynamics of the odd and even number of
flavours. We further determine the critical chemical potential as function of
the $\theta$ angle separating the normal from the superfluid phase of the
theory. Our results will guide numerical simulations and novel tests of the
model's dynamics. The results are also expected to better inform
phenomenological applications of the model ranging from composite Higgs physics
to strongly interacting massive dark matter models featuring number changing
interactions. In the companion work \cite{PartII} we repurpose and upgrade the
approach to determine the impact of the $\theta$-angle and axion physics on
non-perturbative near conformal dynamics related to the fixed baryon charge
sector.
