Short answers, pulled from the story.
Philip Henslowe's diary entry from the 24th of January 1594 records a performance of Titus Andronicus by Sussex's Men at The Rose theatre. Subsequent performances occurred on the 29th of January and the 6th of February of that same year.
Thomas Middleton likely added the fly-killing scene after Shakespeare died in 1616. This scene appears in the First Folio text of 1623 but is found in none of the quarto editions.
Shakespeare drew character names from multiple historical and literary sources including Plutarch's Life of Scipio Africanus for Caius Demetrius Marcus Martius Quintus and Sempronius. Lucius may be named after Saint Lucius or Lucius Junius Brutus founder of the Roman Republic while Lavinia takes her name from Virgil's Aeneid.
The earliest known printed version appeared later in 1594 under the title The Most Lamentable Romaine Tragedie of Titus Andronicus. John Danter entered A booke intitled a Noble Roman Historye of Tytus Andronicus into the Stationers Register on the 6th of February 1594.
Edward Ravenscroft wrote in 1687 that the original play is the most incorrect and indigested piece in all his works seeming rather a heap of rubbish than a structure. Samuel Johnson questioned the possibility of staging the play in 1765 pointing out that the barbarity of the spectacles can scarcely be conceived tolerable to any audience.