Short answers, pulled from the story.
Each Polybian legion contained three hundred horsemen divided into ten turmae of thirty men each. Squadron members elected three officers called decuriones, with the first chosen acting as squadron leader and the other two as deputies. No overall commander is attested for a legion's entire cavalry contingent.
Republican cavalry wore variants of the Corinthian helmet, bronze breastplates or mail cuirasses, and greaves. They carried a lance, the long spatha sword, and sometimes a short throwing spear. By around 150 BC, mail armour had been widely adopted, as confirmed by Polybius and depicted on the monument Lucius Aemilius Paullus erected at Delphi after the Battle of Pydna in 168 BC.
Republican cavalrymen were paid one drachma per day, triple the infantry rate. Their maximum service obligation was ten campaigning seasons, compared to sixteen seasons for infantry.
Citizen cavalry vanished by the end of the first century BC due to a shrinking pool of eligible equites and First Class commoners, the dismantling of the Italian confederate alae after the Social War of 91-87 BC, and the ready availability of superior native cavalry from Gaul, Spain, Thrace, and Numidia at lower cost than Roman citizens.
Roman cavalry suffered crushing defeats at Trebia and Cannae during Hannibal's invasion of Italy from 218 to 216 BC. Philip Sidnell argues these losses stemmed from Hannibal's consistent numerical superiority in horse and from the mismatch between Rome's shock melee cavalry and the nimble hit-and-run tactics of Hannibal's Numidian light cavalry.
Roman cavalry did not use stirrups. The device was introduced to Europe by invading tribes after the collapse of the Western Roman Empire, though the specific tribe responsible is not known. The Roman saddle was a four-horn solid-treed design first used as early as the first century BC.