Also, how far could Southern states and their municipal governments go in suppressing civil rights?
Presumably, you are referring to the use of fire-hoses by Birmingham Police in 1963 against Martin Luther King's "Children's Crusade" as well as other protests in Birmingham. The reaction by Americans the country over was one of shock and horror as images from the protests in Birmingham came to the small screen. In addition to the outrage experienced by the American people, the racial violence worried the Kennedy Administration to the point that they sent in Burke Marshall, head of the Civil Rights Division of the Department of Justice to talk King into ending the protests.
In Sweet Land of Liberty Sugrue writes:
Civil Rights groups throughout the North staged rallies to express solidarity with the Birmingham protesters and their outrage at the police violence. Thousands took to the streets... northern activists held mass meetings to raise bail money and other funds for the Birmingham Protesters."
Not only did the violence of Birmingham affect black activists, but white journalists as well. In its May 24, 1963 Editorial, Life Magazine spoke out for Civil Rights "The...negro needs to have his essential human dignity confirmed... It is the clear obligation of white Americans not to impede this process, but to hasten its fulfillment."
In Parting the Waters Branch writes:
"The morning New York Times featured three of them stacked two columns wide on the front page: on the bottom, state troopers dragging the CORE-SNCC “freedom walkers” to jail in Fort Payne, Alabama; in the middle, Birmingham firemen straining to aim a monitor gun at demonstrators; on top, the police dog sinking its teeth into Walter Gadsden’s midriff. The visual power of the Gadsden photograph was so profound that President Kennedy, like millions of readers, could see nothing else. The picture made him “sick,” he told a morning audience of ADA liberals at the White House."
As you can tell, the violence at Birmingham mortified the American populace, black and white and convinced many of those still sitting on the fence that something had to be done. In regards to the use of municipal forces (policemen, firemen) Southern towns and cities could go as far as they felt necessary to an extent. Bull Connor was able to use his gubernatorial powers to use the police and firemen to harass protesters and activists. But in 1957 when Governor of Arkansas Orval Faubus defied the order to integrate Little Rock High School, President Eisenhower asserted Federal authority and nationalized the Arkansas National Guard.
Sources
Sweet Land of Liberty: The Forgotten Struggle for Civil rights in the North Thomas J. Sugrue
But for Birmingham : The Local and National Movements in the Civil Rights Struggle Glenn T. Eskew
Parting the Waters: America in the King Years 1954-1963 Taylor Branch