Big shift in Claire's attitude! In the past, she displayed little but contempt for "commoners" and arrogant pride in the (supposed) superiority of her class. Here, however, she quickly comes to believe bourgeois-born Matt's side of the story, understanding that the lenient sentence handed down to Dede, a noble, is not only unjust, but "horrible".
This class struggle thing is kinda wack ngl, not enough historical exposition or matters of literacy. If the common folk aren’t educated as said earlier, how are they organizing and spreading word properly? I mean, literacy rates should be low.
Also where is the censorship? Surely the government doesn’t have freedom of press. Why doesn’t the government do a purge? This is what the nobles would reasonably do.
The king is said to have recently instituted reforms in an attempt to nudge the kingdom towards meritocracy. Therefore, prior to this action against the academy, he's probably been hesitant to crack down. Remember, too, that shortly before the attack, the princes estimated that the peasant rights movement had drawn only about 20 people, including the curious but uncommitted.
As to your first question, I imagine that word has been spread the old-fashioned way: in private conversations and small gatherings (in pubs, at work, after soapbox speeches, etc).