Short answers, pulled from the story.
Construction of the Tuileries Palace began in 1564, commissioned by Queen Catherine de' Medici after the death of her husband, Henri II, in 1559. Philibert de l'Orme served as the original chief architect until his death in 1570, when Jean Bullant took over.
The Tuileries Palace was set on fire on the 23rd of May 1871 by twelve men acting under Jules Bergeret during the suppression of the Paris Commune. They used petroleum, liquid tar, and turpentine, and the fire burned for 48 hours, gutting the building down to its foundations.
Demolition of the Tuileries Palace began in February 1883 and was completed on the 30th of September 1883, eleven years after the 1871 fire had left it in ruins. The stone walls had survived the fire intact, but in 1882 officials ordered them torn down anyway.
The Tuileries Palace served as the Parisian residence of most French monarchs from Henri IV to Napoleon III, including Louis XIV, Louis XV, Louis XVI, Napoleon Bonaparte, Louis Philippe d'Orléans, and Louis-Napoléon Bonaparte, who ruled as Napoleon III. Louis XIV lived there only briefly before moving the court to Versailles.
No, the Tuileries Palace was never rebuilt. In 2003 the Committee for the Reconstruction of the Tuileries proposed reconstructing it, and a 2006 estimate put the cost at 300 million euros, but in 2008 Michel Clément, Director of Architecture and Heritage, said reconstruction was not a priority.
After the Tuileries Palace was demolished, the courtyard that once faced it remained open to the west, forming an elevated terrace between the Place du Carrousel and the Tuileries Garden. This open space also connects to the Axe historique, the sightline running through the Place de la Concorde and the Arc de Triomphe to La Défense.