Short answers, pulled from the story.
Titus Andronicus is believed to have been written between 1588 and 1593. The earliest definite recorded performance is noted in Philip Henslowe's diary on the 24th of January 1594, which established 1593 as the latest possible date of composition. Some scholars, including E. A. J. Honigmann and Alan Hughes, have argued for an earlier date, possibly as early as 1586 or 1588.
The primary source for the rape and mutilation of Lavinia, and Titus's subsequent revenge, is Ovid's Metamorphoses, written around 8 AD, which is actually referenced within the play itself. Shakespeare also drew on Seneca's Thyestes for the revenge feast where a parent unknowingly eats their own children, and on Livy's Ab urbe condita for the story of Verginia, which shapes the moment Titus kills Lavinia.
The question of co-authorship remains contested. Multiple scholars using different methods, including rhetorical analysis, rare-word tests, and metrical analysis, have concluded that George Peele wrote Act 1, sections 2.1 and 4.1. The New Oxford Shakespeare proposed that the Fly Scene (Act 3, Scene 2) was a late addition by Thomas Middleton. Some scholars, including Jonathan Bate and Eugene M. Waith in their respective editions, have argued Shakespeare wrote the play alone.
Critics objected primarily to the play's extreme graphic violence, including rape, mutilation, and cannibalism. T. S. Eliot called it one of the stupidest and most uninspired plays ever written in 1927, and John Dover Wilson in 1948 wrote that it seemed to bump along like a broken-down cart laden with bleeding corpses. The Victorian era largely refused to stage it on grounds of barbarity.
Marcus's speech upon discovering the mutilated Lavinia is one of the most debated passages in the play. Critics including John Dover Wilson called it an aesthetic failure, and Peter Brook cut it entirely from his 1955 RSC production. Defenders such as Jonathan Bate argue the ornate language attempts to restore figuratively what has been taken from Lavinia, while the 1987 RSC production at The Swan, directed by Deborah Warner, demonstrated that the speech can be deeply moving when delivered in performance.
Titus Andronicus was extremely popular when first performed. Philip Henslowe recorded that the performance on the 24th of January 1594 earned three pounds eight shillings, making it the most profitable play of the season. Subsequent performances followed on the 29th of January and the 6th of February. Its popularity declined significantly over the course of the seventeenth century.