Short answers, pulled from the story.
Romulus established the Roman cavalry regiment of 300 men called the Celeres as his personal escort during the early founding period of Rome. This royal cavalry regiment was later doubled in size to 600 men by King Tarquinius Priscus between 616 and 578 BC.
Equites were required to serve up to 10 years of service in the cavalry between the ages of 17 and 46. They received a drachma per day which was triple the infantry rate and served a maximum of ten campaigning seasons compared to sixteen for the infantry.
The earliest extant representations of Roman cavalrymen appear on coins dated to the era of the Second Punic War between 218 and 201 BC showing Corinthian helmets and small round shields. Mail had certainly been adopted by ca. 150 BC as Polybius states that the First Class were expected to provide themselves with mail cuirasses.
The Roman cavalry failed against Hannibal's nimble Numidian light cavalry whose hit and run tactics exasperated them. These losses were rendered possible because of a powerful Carthaginian cavalry force that defeated Roman forces at battles like Trebia and Cannae.
By the end of the 1st century BC citizen cavalry disappeared completely from the Roman army and was replaced by foreign auxiliaries. The Jugurthine War is the last war in which Roman and Italian confederate cavalry is attested as having played a significant part before this transition.
Augustus restored to each Roman legion a small citizen cavalry force of 120 men recruited from the legionaries themselves. He also made a regular Auxilia corps of non-citizen soldiers who were subjects recruited from provinces controlled by Rome with strong native cavalry traditions.