A major part of the civil rights movement was the ability of local and national organizations to successfully gain the attention of media outlets, which, in turn, garnered sympathy for the movement and the plight of African Americans suffering under the explicit racism of Jim Crow and the actions of men like Eugene "Bull" Connor. The incidents that found their way onto mainstream national headlines did not happen on accident, organizations debated over what issues or events took precedent over others, and carefully constructed a narrative that might be more appealing to a wide audience. I recommend Danielle McGuire's At the Dark End of the Street: Black Women, Rape, and Resistance- A New History of the Civil Rights Movement From Rosa Parks to the Rise of Black Power. McGuire covers the creation of committee groups in response to incidents where black women were raped by white men with impunity. These coalition groups networked with other organizations to garner the attention of national media outlets. Perhaps the most interesting part of McGuire's work is her coverage of Rosa Parks and the Montgomery Bus Boycott, which Parks and other local women were almost entirely responsible for, but Martin Luther King jr. received the most attention due to media attitudes and a movement within the black church to make the entire civil rights movement appear to originate from the work of black pastors. I also recommend Mary Dudziak's Cold War Civil Rights: Race and the Image of American Democracy. This is a classic, and the coverage internationally over the Little Rock Nine represents the importance of press to the civil rights movement.

I don't have immediate answers to your questions about the general attitudes or biases of the media, but hopefully these two sources will give you something to look at for answers.