Short answers, pulled from the story.
Florence Cathedral was substantially completed in 1436 when its dome was finished; Filippo Brunelleschi engineered and built the dome, with construction of the dome running from 1420 to 1436. The full project, including Verrocchio's copper ball atop the lantern, was not finished until 1469, having begun in 1296 under architect Arnolfo di Cambio.
The Florence Cathedral dome is still the largest masonry dome ever constructed, spanning 44 metres and rising to a height of 114.5 metres. It was the first octagonal dome in history built without a temporary wooden supporting frame, a feat Brunelleschi achieved through an innovative double-shell design, internal stone-and-iron chain hoops, herringbone brickwork, and specially invented hoisting machines.
During the consecration of Florence Cathedral on the 25th of March 1436, the motet Nuper rosarum flores by Guillaume Dufay was performed. The consecration was conducted by Pope Eugene IV.
Lorenzo Ghiberti was the main competitor against Filippo Brunelleschi in the 1418 competition announced by the Arte della Lana to design and build the dome. Brunelleschi won and was later supported by Cosimo de' Medici; Ghiberti was appointed coadjutor and paid an equal salary, though he spent most of his time on other projects.
Yes, cracks in the Florence Cathedral dome were recorded as early as 1639 and a 1984 survey counted 493 cracks of various sizes. The most serious, Type A cracks, run through both shells and measure up to 55-60 millimetres wide in some sections; however, a digital monitoring system installed in 1987 shows the major cracks widening at approximately 3 millimetres per century, and to date they have not caused catastrophic damage.
Leonardo da Vinci was a young apprentice in the workshop of sculptor Andrea del Verrocchio when Verrocchio was commissioned to hoist the gilt copper ball atop the cathedral's lantern in 1469. Fascinated by Brunelleschi's hoisting machines, which Verrocchio used for the task, Leonardo made a series of sketches of them and noted in the G manuscript of Paris: 'Remember the way we soldered the ball of Santa Maria del Fiore'.