Short answers, pulled from the story.
Achilles was the son of Thetis, a sea-nymph and Nereid, and Peleus, king of the Myrmidons in Phthia. Zeus and Poseidon had both wanted to marry Thetis until a prophecy, originally spoken by Themis, warned that her son would surpass his father, after which both gods withdrew and arranged her marriage to Peleus instead.
The name Achilles combines the Greek words for 'distress or grief' and 'people or soldiers', yielding the sense of 'he whose people have distress' or 'he who has the people distressed'. Scholars including Gregory Nagy, following Leonard Palmer, have noted that this name creates an ironic tension with Achilles's reputation as the hero of glory in war.
No. The invulnerable heel does not appear in Homer's Iliad. Homer actually describes Achilles being wounded by a spear that grazes his elbow in Book 21. The first written account of the heel vulnerability appears in the Achilleid, an unfinished epic by Statius composed in the first century CE, which describes Thetis dipping the infant Achilles in the river Styx while holding him by his left heel.
Achilles withdrew from battle after Agamemnon, the Greek commander, took Briseis, the daughter of Briseus, who was Achilles's slave and someone he loved. Agamemnon had been forced to give up his own slave and compensated himself by seizing Briseis, which Achilles regarded as a humiliation. He returned to battle only after Hector killed his companion Patroclus.
Ancient sources give differing accounts. The most common tradition holds that Paris shot him with an arrow, guided by Apollo, near the end of the Trojan War. Statius added that the arrow struck his heel. Earlier Epic Cycle fragments, including the works of Lesches of Pyrrha and Arctinus of Miletus, mention no heel vulnerability, and later vase paintings often show arrows striking his torso rather than his heel.
Achilles was venerated across a wide area, including sanctuaries in Sparta, Elis, Thessaly, and the Magna Graecia cities of Tarentum, Locri, and Croton. The most prominent cult center was an island in the Black Sea now identified with Snake Island near Kiliia, Ukraine, where a temple measuring thirty meters to a side once stood. Dedicatory inscriptions from the sixth century BCE onward attest the cult there, and by the third century CE the city of Olbia called Achilles 'lord of the sea' and honored him alongside the Olympian gods.