The musical Chess originated as a concept album in 1984 and made its stage debut in London in 1986. The music was written by Benny Andersson and Björn Ulvaeus of ABBA fame, and the lyrics were written by Tim Rice.
The musical primarily uses chess as a plot device and as a metaphor for the Cold War rivalry between the United States and Soviet Union. However, one song in the production called "The Story of Chess" purports to be a full abbreviated history of the board game from its invention to modern times. This song is usually the first song in the production and performed by an ensemble of pseudo-narrator characters (although this varies from staging to staging).
The full lyrics are here, and a recording from 2008 can be heard here. In summary, the claims the song makes can be summarized as follows:
How accurate is this account? Are there any significant parts of the real history that are left out or glossed over? And when did the game become something we would recognize as at least resembling modern chess? I've seen a fair bit of conflicting information online, so some suggested sources for further reading would also be helpful.
My most concise history of chess can be found in an answer to a previous question. That will answer some of your questions, but let me specifically respond to the claims you list:
True--We believe chess came from India in the early 6th century.
Unknown--We don't know what inspired the creation of chess.
False--Chess spread to China first, then to Persia. That said, the Muslim conquest of Persia and later North Africa would lead to introduction into Russia and Spain.
False--The Sasanian Empire and Umayyad Caliphate did not change much. We owe most of the standardization and many of the modern rules to late 15th century Spain.
True--Chess was popular in the Byzantine Empire, who introduced chess into Greece and other areas. However, chess figures would have been too costly for most citizens of the Byzantine Empire.
False--Chess had spread throughout Europe well before the Renaissance--consider the Lewis Chessmen, which were made in Norway in the 12th century. However, chess did dramatically rise in popularity during the Renaissance.
True--See 4.
If you'd like to read a book on the topic of the musical you reference, I'd suggest reading an interesting collaboration between a Grandmaster who fled to the West, a former KGB agent, and a Russian historian:
Gulko, Boris. The KGB Plays Chess: The Soviet Secret Police and the Fight for the World Chess Crown. Russell, 2010.