Short answers, pulled from the story.
Baroque architecture began in the late 1500s when Rome became the stage for a new kind of architecture designed to fight a spiritual war. The Catholic Church needed a visual weapon against the rising tide of Protestantism and turned to architects who could create spaces that inspired astonishment and awe.
Pope Urban VIII appointed Gian Lorenzo Bernini as chief Papal architect after Carlo Maderno died in 1629. Bernini transformed Rome into an enormous theater by designing squares, fountains, and interiors including the Fontana del Tritone and the baldacchino inside St. Peter's Basilica.
Jules Hardouin-Mansart constructed the Hall of Mirrors at the Palace of Versailles between 1668 and 1678. This building stands alongside Claude Perrault's classical variant of Baroque for the Louvre Palace east wing which began construction in 1664.
A distinct Ottoman Baroque style emerged in Istanbul during the 1740s to replace earlier classical designs. Mahmud I began the Nuruosmaniye Mosque complex in October 1748 which Osman III completed by December 1755.
Baroque architects used Quadratura combined trompe-l'œil painting with sculpture to create illusions of looking up into heaven. They employed Chiaroscuro strong contrasts of darkness and light for dramatic effect and designed elliptical spaces that eliminated right angles.