Short answers, pulled from the story.
JPL Horizons provides access to Solar System data and generates highly accurate ephemerides for Solar System objects. Scientists and mission planners use it to determine precise positions of planets, asteroids, comets, and other Solar System bodies at any point in time.
JPL Horizons integrates the equations of motion in Cartesian coordinates (x, y, z) rather than relying on classical orbital elements such as periods and eccentricities. It accounts for gravitational perturbations from all planets, several large asteroids, and additional small physical forces, then adjusts initial conditions to fit modern astrometric measurements.
Horizons switched to the DE440/441 solution during the week of the 12th of April 2021, replacing the DE430/431 ephemeris that had been in use since August 2013. The new solution added seven years of additional astrometric data and significantly improved the model for Jupiter, Saturn, Pluto, and the Kuiper Belt.
Objects on outbound ejection trajectories, such as C/1980 E1, show an eccentricity greater than 1, and Horizons represents their apoapsis distance and orbital period as 9.99E+99. This indicates that the object is not gravitationally bound and is leaving the Solar System.
JPL Horizons can be accessed via a web interface, email, or telnet. The web interface provides partial access, while email and telnet both offer full access. All three methods can be automated for programmatic use.
JPL began transitioning Horizons from a common gateway interface (CGI) to an application programming interface (API) in September 2021. The change makes it easier to integrate Horizons queries into scientific software pipelines and automated tools.