Short answers, pulled from the story.
John Fletcher and William Shakespeare co-wrote the play. John Fletcher wrote the prologue and most of Act Two through Act Four, while William Shakespeare handled scenes one through three of Act One and parts of Act Five.
The play was entered into the Stationers Register on the 8th of April 1634. Bookseller John Waterson published the quarto later that year with printer Thomas Cotes producing physical copies for distribution.
Palamon and Arcite fought because they both fell instantly in love with Princess Emilia after seeing her from their prison cell window. Their friendship shatters into bitter rivalry over a woman they have never spoken to.
The jailer's daughter went mad after being rejected by Palamon. Her father tried to restore her mind with the help of a doctor who brought a wooer to pretend he is the prisoner she loves.
A copy discovered in Salamanca, Spain, dates back to the early seventeenth century. This edition may be the oldest surviving version of any Shakespeare work in Spanish-speaking lands.