Short answers, pulled from the story.
Mars was usually considered the son of Jupiter and Juno. In Ovid's account in the Fasti, however, Juno conceived Mars alone, using a magic flower obtained from the goddess Flora, after Jupiter had given birth to Minerva without a female partner.
Mars functioned as an agricultural guardian because his role was to ward off hostile forces threatening crops and livestock, a protective function inseparable from his warrior nature. The Arval Brothers and Cato both invoked Mars to defend harvests from disease, poor weather, and other misfortune. The Roman farming and military campaigning seasons coincided, reinforcing his dual role.
The Temple of Mars Ultor, dedicated in 2 BCE in the Forum of Augustus, honored Mars as the Avenger. Augustus built it to mark his defeat of Caesar's assassins at Philippi in 42 BCE and the recovery of Roman battle standards lost to the Parthians at Carrhae in 53 BCE. The Senate was required to meet there when deliberating war and peace.
The woodpecker and the wolf were the wild animals most sacred to Mars. Plutarch explains the woodpecker's association by noting its aggressive courage and the strength of its beak. The wolf is familiar from the myth of the she-wolf who suckled Romulus and Remus. In Roman Gaul, the goose was also linked to Celtic forms of Mars because of its easily provoked aggression.
Mars gave his name to the month of March, originally the first month of the Roman calendar. The Latin adjective mārtiālis produced the English word "martial." Personal names including Marcus, Mark, and Martin all derive from his name. In Romance languages, Tuesday is named after Mars: martes in Spanish, mardi in French, martedì in Italian.
In Gaul, Roman Spain, and Britain, Mars was frequently identified with local Celtic deities and invoked as a healer. His battlefield ability to repel enemies was applied to illness. Mars Lenus had a major healing cult at the capital of the Treveri in present-day Trier, and Mars Vorocius was worshipped as a curer of eye afflictions at a curative spring shrine at Vichy.