  We introduce a convenient set of analytical tools (the Gaussian formalism)
and diagrams (symplectic circuits) to analyze multi-mode scattering events in
analog gravity, such as pair-creation a l\'a Hawking by black hole and white
hole analog event horizons. The diagrams prove to be valuable ansatzes for the
scattering dynamics, especially in settings where direct analytic results are
not straightforward and one must instead rely on numerical simulations. We use
these tools to investigate entanglement generation in single- and multi-horizon
scenarios, in particular when the Hawking process is stimulated with classical
(e.g., thermal noise) and non-classical (e.g., single-mode squeezed vacuum)
input states -- demonstrating, for instance, that initial squeezing can enhance
the production of entanglement and overcome the deleterious effects that
initial thermal fluctuations have on the output entanglement. To make further
contact with practical matters, we examine how attenuation degrades quantum
correlations between Hawking pairs. The techniques that we employ are generally
applicable to analog gravity setups of (Gaussian) bosonic quantum systems, such
as analog horizons produced in optical analogs and in Bose-Einstein
condensates, and should be of great utility in these domains. We show the
applicability of these techniques by putting them in action for an optical
system containing a pair white-black hole analog, extending our previous
analysis of [Phys. Rev. Lett. 128, 091301 (2022)].
