Short answers, pulled from the story.
The Roman Senate is traditionally said to have been founded by Rome's first king, Romulus, initially with 100 members. Rome was traditionally founded in 753 BC, and the Senate dates to those earliest days of the city.
During the Republic, the Roman Senate controlled the state treasury and could authorize all disbursements of public funds, directed the consuls in military campaigns, supervised provincial administration, oversaw judicial proceedings in serious criminal cases, managed Rome's diplomatic relations, and issued decrees called senatus consulta that were almost always obeyed in practice even though they lacked the force of law.
Under the emperors, the Roman Senate lost real governing authority even as it retained formal legal standing. Augustus reduced its membership from 900 to 600. The emperor Tiberius transferred electoral powers to the Senate but required imperial approval for all candidates, making senatorial elections largely ceremonial. Senatorial decrees did gain the full force of law during the imperial period, replacing the popular assemblies, but the emperor's control over the body was absolute.
The exact date is uncertain, but the Western Roman Senate likely ceased to function as a real institution shortly after 603, the date of its last recorded public act. Pope Gregory I lamented the near disappearance of the senatorial order in a sermon in 593. In 630, Pope Honorius I converted the Senate's meeting house, the Curia Julia, into a church called Sant'Adriano al Foro.
The Eastern Roman Senate in Constantinople survived until at least the beginning of the 13th century. Its last known act was the election of Nicholas Kanabos as emperor in 1204 during the Fourth Crusade.
No. Although the Latin word Senatus is the etymological origin of the modern term senate, the Roman Senate was not a de jure legislative body and was not the direct ancestor or predecessor of modern parliamentary or senatorial structures.