Short answers, pulled from the story.
Modern archaeologists identify Castel Gandolfo on a sunlit ridge overlooking Lake Albano as the most likely site of ancient Alba Longa. Livy placed the town between the Alban Mount and the Alban Lake, sparking centuries of debate about its true position.
Ascanius built Alba Longa on the slope of Mount Alba thirty years after Lavinium's founding. He resettled six hundred families there as a colony of Lavinium while his descendants ruled the Latins for another five hundred years.
Roman king Tullus Hostilius succeeded Numa Pompilius in the seventh century BC and seized upon mutual accusations of robbery as pretext for conflict with Albans. The Albans became substance vassal states of Rome after the duel concluded following King Cluilius death from unspecified causes.
Worship of Vesta began in Lavinium mother-city of Alba Longa before transferring to Alba Longa and then reaching Rome. A shrine to Jupiter Latiaris stood on Monte Cavo or Mons Albanus where annual Feriae Latinae gatherings occurred under Alba's aegis.
Alba Longa reportedly led roughly thirty cities within the Latin League during its peak influence. Thirty held great importance among Latin kingdoms just as twelve did for Ionians, creating a complex network of alliances across central Italy.