Publishing military secrets is legal (eg https://en.wikipedia.org/wiki/United_States_v._Progressive,_....). Journalists don't have a duty to keep classified information secret, only the people who've agreed to keep it secret do.
Obviously false, see Assange.
Assange hasn't been convicted of anything yet. Seeing as he was part of a conspiracy to steal the classified information, probably will be though.
The bit of the Espionage Act that conflicts with my previous post is unconstitutional.
Otherwise, the rest of censorship comes from social pressure; or someone with hurt feelings trying to twist the courts to enforce their will.