> I would not be surprised if Death Star construction was initiated during the Old Republic.
It actually was. In the prequel trilogy, we see that the Separatists have plans for it, which are then acquired by the Empire at the end of the film.
> The first Death Star's schematics are visible in the scenes on Geonosis in Star Wars: Episode II – Attack of the Clones showcasing the early development of the Death Star prototype, the Death Star plans were designed by Geonosians led by Archduke Poggle the Lesser, a member of the Confederacy of Independent Systems, and is shown early in construction at the end of Star Wars: Episode III – Revenge of the Sith.