Some have been calling for the firing of people who publicly praised the murder of Charlie Kirk, or at least who argued that the murder was justified or defensible. I'm not wild about such calls; I think we generally need less cancel culture, not more, even as to people who say morally repugnant things. (Among other things, these calls for firing tend to spiral, to cover a wide range of other speech beyond the outrageous statements that first led to them.) But here let me focus not on the ethical or pragmatic question, but the legal one: If a government employer fires an employee for such speech, would that violate the employee's First Amendment rights?
[1.] Praising violence doesn't generally fall into any existing First Amendment exception, so that means it's protected against the government as sovereign—against criminal punishment, civil liability, and the like. The "incitement" exception is limited to speech that is intended to and likely to cause imminent illegal conduct. Praise of a past murder wouldn't qualify: Even if such praise may have a long-term tendency to influence people in the future to do bad things, the Court has rejected this "bad tendency" test for punishing speech.
Intentionally soliciting a criminal attack on a particular person may potentially be punishable as "solicitation" of crime, under U.S. v. Williams (2008) and U.S. v. Hansen (2023). But that certainly wouldn't apply to mere praise or justification of an attack that had already happened. (Just when it would apply to general advocacy of a future attack is a complex and unsettled question.) For more on these rules, see this post on the Graham Linehan controversy.