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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 lá 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)].