Short answers, pulled from the story.
The lyrics were written by Józef Wybicki in Reggio Emilia, northern Italy, between the 16th and the 19th of July 1797. He composed the poem while the Polish Legions were quartered in the city, serving with Napoleon Bonaparte's army.
Poland had been partitioned out of existence by Russia, Prussia, and Austria in 1795. Polish patriots formed military legions within the French Revolutionary Army, and Wybicki wrote the song to raise the morale of Jan Henryk Dąbrowski's Polish Legions then stationed in Reggio Emilia. The song expressed the belief that the Polish nation survived as long as Polish people fought in its name.
The song became the de facto national anthem when Poland declared independence as the Second Polish Republic in 1918. It was officially adopted by law in 1927.
The composer is unknown. The melody is most likely Wybicki's adaptation of a folk tune popular in the second half of the 18th century. It was long misattributed to Michał Kleofas Ogiński, but his sheet music was discovered in 1938 and proved to be a different composition. The current official score is arranged by Kazimierz Sikorski.
Poland Is Not Yet Lost directly inspired the Ukrainian anthem "Ukraine Is Not Yet Perished", the Israeli anthem "Hatikvah", the Croatian reveille "Croatia has not yet fallen", and the Slovak and Yugoslav anthem "Hey, Slavs." The Slovak poet Samo Tomášik wrote "Hej, Sloveni" to a slowed version of the Polish melody during the European Revolutions of 1848.
The original manuscript remained in the hands of Wybicki's descendants until February 1944, when it was lost in the Charlottenburg home of his great-great-grandson, Johann von Roznowski, during Allied bombing of Berlin. The manuscript survives today only through twenty-four facsimile copies made in 1886 by Wybicki's grandson Edward Rożnowski, who donated them to Polish libraries.
At the 1945 UN inauguration concert in San Francisco, pianist Artur Rubinstein protested the absence of a Polish delegation by addressing the audience and then playing "Poland Is Not Yet Lost" loudly and slowly at the piano, repeating the final section in a thunderous forte. The audience rose and gave him a standing ovation.