Short answers, pulled from the story.
Ivan the Terrible ordered Saint Basil's Cathedral built between 1555 and 1561 to commemorate his military victories over the Kazan and Astrakhan khanates. Placing it outside the Kremlin walls rather than inside was a deliberate political statement in favour of posad commoners and against hereditary boyars.
The architect's identity has never been definitively established. Tradition names two builders, Barma and Postnik, and researchers have proposed that both names refer to a single person, Postnik Yakovlev. The official Russian cultural heritage register lists "Barma and Postnik Yakovlev."
Most historians regard the blinding story as a myth. Postnik Yakovlev, the most likely candidate for the architect, is documented as active at least through the 1560s and worked on the Cathedral of the Annunciation in Moscow and on the walls and towers of the Kazan Kremlin after completing Saint Basil's. The legend most likely originated with a separate account of Ivan III blinding the architect of the fortress of Ivangorod.
Saint Basil's Cathedral contains ten individual chapels. The original structure comprised nine churches arranged around a central core; a tenth chapel was added in 1588 over the grave of the local saint Vasily (Basil). The largest, the central Church of the Intercession, rises 46 metres internally but has a floor area of only 64 square metres.
Yes. In the first half of the 1930s, Lazar Kaganovich's reconstruction plans for Moscow made the cathedral an obstacle. Preservationist Pyotr Baranovsky was arrested after protesting a scheduled demolition, and in the autumn of 1933 the building was struck from the heritage register. By 1937 even hard-line Bolshevik planners accepted that the building should be spared.
Saint Basil's Cathedral became a public museum in 1923, though religious services continued inside until 1929, when it was completely secularized. It has operated as a division of the State Historical Museum since 1928 and became part of the Moscow Kremlin and Red Square UNESCO World Heritage Site in 1990.