That's where you're wrong. Those school receive government monies so by extension they represent the government. When they invite someone to speak, then prevent that speech they are violating the First Amendment. They are PUBLIC institutions.
Not really. The government is contracting them for a service... that service being to provide education for students or conduct research. That does not require them to provide a forum for anyone...
If a student group invites a guest to speak and the school agrees to the contract the school and the speaker, they are contractually obliged to provide a forum. Which is what was done between Berkley and Ann Coulter. They are also obliged to provide adequate security for said forum. Berkley is in breach of contract if they later refuse her a forum.
Berkley, itself, is not in violation of the First Amendment unless they are trying to stifle Ann and prevent their students from hearing her speak, which doesn't appear to be the case. Berkley tried to arbitrarily set another date for her to speak, but they scheduled it when nobody was in school (sound kind of fishy to me).
The next question would be: Is Berkley's refusal to supply adequate security for Ann come from not wanting her to speak, or fear and cowardice in the face of Fascists/Anarchists threats of violence?
(Yes, I've altered my opinion on this issue.)