Short answers, pulled from the story.
Woodward argued that racial segregation in the American South was a late political invention, not an inevitable outgrowth of Southern tradition. Jim Crow laws, he showed, were largely imposed in the 1890s, decades after Reconstruction ended, following a period in the 1880s when some informal racial separation coexisted with what he called "forgotten alternatives."
The attribution is approximate. King cited The Strange Career of Jim Crow in a speech in Montgomery, Alabama on the 23rd of March 1956, and said it proved racial segregation was, in his own words, "a political stratagem" rather than a natural condition. The phrase "historical Bible of the Civil Rights Movement" is a popular summary that does not match what King said.
Woodward won the Pulitzer Prize for History in 1982 for Mary Chesnut's Civil War, his edited version of the Civil War diary of Mary Chesnut. He also won the Bancroft Prize for Origins of the New South, 1877-1913.
Woodward taught at Johns Hopkins University from 1946 to 1961, then served as Sterling Professor of History at Yale from 1961 to 1977. He directed 25 PhD dissertations; his students included James M. McPherson of Princeton, Patricia Nelson Limerick of the University of Colorado at Boulder, Steven Hahn of New York University, and Barbara Fields of Columbia.
In April 1974, John Doar of the U.S. House Judiciary Committee asked Woodward to provide historical context for the impeachment process against President Richard Nixon. Woodward and fourteen other historians completed a 400-page report titled "Responses of the Presidents to Charges of Misconduct" by late June 1974. Nixon resigned before the report could be used, and it entered the public domain; it was republished in 2019 as "Presidential Misconduct."
Woodward began his career on the political left in the 1930s and was a leading liberal supporter of civil rights by the 1950s. After clashes with New Left historians in the late 1960s, he moved steadily rightward, opposing affirmative action, multiculturalism, and academic theory. By 1987 he had joined the National Association of Scholars, a group explicitly opposed to the academic left.