Is hate speech protected under the first amendment?

[hissingfight224]

Depending on the situation they could be arrested for harassment.
Yes, a suspension is possible. Probably not for first offenses perhaps, but a pattern of the behavior.
Fired, certainly depending on the person's job. A trucker isn't going to be treated the same as a teacher for example.

As the guy below me seems to ignore, the questions posed have to do with employers and schools. The national government cannot censor your speech (actually there are exceptions) , however your employer or school officials do reserve the right to take disciplinary actions if one breaks their institutional rules or regulations.