Short answers, pulled from the story.
Scholars place Quintus Smyrnaeus in the latter part of the 4th century AD, though some researchers argue for the 3rd or even the 2nd century. Evidence from book six suggests he wrote before Emperor Theodosius I suppressed man-and-beast fights between 379 and 395 AD.
The narrative begins after Hector's body was regained by the Trojans at the end of Homer's Iliad and covers events including the arrival of Amazon queen Penthesileia and the death of King Memnon. It continues with Achilles' funeral games, the contest for his arms resulting in Aias taking his own life, and concludes with the building of the wooden horse leading to the destruction of Troy.
Quintus modeled his epic on Homer while drawing heavily from Arctinus of Miletus' Aethiopis and Iliupersis as well as Lesches' Ilias Mikra. He also incorporated elements from Greek tragedy such as Sophocles' depiction of Ajax into books five through twelve.
Cardinal Bessarion found the only known manuscript of the poem at Otranto in Calabria during 1450 under the title Quinti Calabri derelictorum ab Homero libri XIV. Lorenz Rhodomann gave him his current name in 1577 when he included a Latin translation by Michael Neander.
A.S. Way produced an English version for the Loeb Classical Library volume 19 in 1913 and Francis Vian published a new Greek text with partial commentary across three volumes between 1963 and 1969. Alan James released his own English translation with newly edited text and commentary alongside detailed work by Kevin Lee on book five.