Short answers, pulled from the story.
The Index Librorum Prohibitorum was a list of publications condemned by the Sacred Congregation of the Index of the Roman Catholic Church, active from 1560 to 1966. Catholics were forbidden to print or read the listed works, subject to the authority of the local bishop. The final edition, published in 1948, contained 4,000 titles.
The Index listed works by Galileo Galilei, René Descartes, Immanuel Kant, John Locke, Voltaire, Victor Hugo, Jean-Paul Sartre, Simone de Beauvoir, Baruch Spinoza, Francis Bacon, Nicolaus Copernicus, Niccolò Machiavelli, Blaise Pascal, David Hume, and Jean-Jacques Rousseau, among many others. The first woman listed was Magdalena Haymairus in 1569.
The Holy Office studied Mein Kampf for three years but decided it should not appear on the Index because Adolf Hitler was a head of state. The decision was justified by citing chapter 13 of Paul the Apostle's Epistle to the Romans regarding state authority coming from God. The Vatican did separately criticize Mein Kampf in the encyclical Mit Brennender Sorge in March 1937.
The Index was effectively dissolved in June 1966, when the Congregation for the Doctrine of the Faith announced it no longer had the force of ecclesiastical positive law with associated penalties. The process began on the 7th of December 1965, when Pope Paul VI reorganized the Holy Office and did not list the Index among its responsibilities.
The Index was directly enforceable only within the Papal States. Elsewhere it applied only when adopted by civil authorities, which occurred in several Italian states. France did not recognize the Church's Index at all, operating its own separate censorship system. Spain maintained its own parallel list called the Index Librorum Prohibitorum et Expurgatorum.
No. Contrary to a widely held belief, Charles Darwin's works were never included in the Index at any stage of its history.