Short answers, pulled from the story.
Troilus and Cressida was probably written in 1602, shortly after the completion of Hamlet. It was entered into the Register of the Stationers' Company on the 7th of February 1603 and published in quarto in two separate editions in 1609.
The nineteenth-century literary critic Frederick S. Boas coined the term "problem play" to describe Troilus and Cressida alongside Measure for Measure and All's Well That Ends Well. He borrowed the phrase from the socially conscious drama of Ibsen and Shaw to describe plays that center on a social or political problem in a way that promotes debate rather than easy resolution.
The love story of Troilus and Cressida is not part of Greek mythology. It originated in medieval literature, first appearing between 1155 and 1160 in the Roman de Troie by Benoit de Sainte-Maure. Shakespeare drew primarily on Chaucer's Troilus and Criseyde (c. 1380), as well as John Lydgate's Troy Book and Caxton's Recuyell of the Historyes of Troye.
The First Folio classed Troilus and Cressida among the tragedies, but its pages are unnumbered, its title is absent from the Table of Contents, and it appears squeezed between the histories and tragedies. Scholars believe it was a very late addition, inserted wherever space allowed. The 1609 Quarto had labeled it a history.
Peter Holland of Cambridge University identified a pattern in which productions of Troilus and Cressida tend to coincide with periods of impending or ongoing conflict. William Poel's 1912 production preceded World War One; Michael Macowan's 1938 modern-dress production at the Westminster Theatre coincided with the Munich crisis; and a 2012 Swan Theatre production resonated with the wars in Iraq and Afghanistan.
A 1956 RADA staging of the play at the Vanbrugh Theatre, adapted from a 1954 BBC broadcast called The Face of Love, provided Albert Finney with his first lead stage role.