Short answers, pulled from the story.
Karl Marx never called himself a Marxist. In 1883, he wrote a letter to his son-in-law Paul Lafargue and French labor leader Jules Guesde, stating that if their politics represented Marxism, then one thing is certain and that is that he is not a Marxist.
The core of Marxism lies in its materialist methodology known as historical materialism. This theory analyzes the underlying causes of societal development from the perspective of the collective ways in which humans make their living, as found in The German Ideology written in 1845 and the preface to A Contribution to the Critique of Political Economy published in 1859.
The October Revolution occurred in 1917 when the Bolsheviks took power from the Russian Provisional Government. This event established the first socialist state based on the ideas of soviet democracy and Leninism, which promised to end Russian involvement in World War I and establish a revolutionary worker's state.
At the turn of the 21st century, China, Cuba, Laos, North Korea, and Vietnam remained the only officially Marxist Leninist states remaining. A Maoist government led by Prachanda was also elected into power in Nepal in 2008 following a long guerrilla struggle.
The term Marxism itself was popularized by Karl Kautsky, who considered himself an orthodox Marxist during the dispute between Marx's orthodox and revisionist followers. Engels did not support using Marxism to describe either Marx's or his views, claiming that the term was being abusively used as a rhetorical qualifier by those attempting to cast themselves as genuine followers of Marx.
Philosopher Karl Popper criticized Marxism in his books The Poverty of Historicism and Conjectures and Refutations, arguing that the theory avoided falsification by adding ad hoc hypotheses and degenerated into pseudoscientific dogma. Austrian economists including Carl Menger, Eugen Böhm von Bawerk, and Ludwig von Mises attacked the law of value and the economic calculation problem, claiming that rational economic activity is impossible in a socialist commonwealth.