In California, parents can be held criminally responsible for their underaged child. Parents can be held legally responsible for their minor children’s actions in both civil and criminal court. According to California Penal Code , if a parent fails to fulfill his or her “duty to exercise reasonable care, supervision, protection, and control over their minor child,” they have committed a misdemeanor crime.
Schools have been using guidelines set forth from the Secret service and the U.S. Dept of Education to discern whether threats are viable or not. Schools themselves do not only act on these threats but alert the local and applicable law enforcement agency. They too assess whether the threat is viable before acting. In this case they involved a judge, who deemed the threat viable enough to issue a search warrant. Unlike what many want us to believe, we do not have kids arrested just for eating their peanut butter and jelly sandwich into the shape of a gun anymore. Not only is a stated/posted threat part of the decision making, but pre-threat behaviors are heavily used. IOW's it took many people, with experience and professional guidance, that decided it was a viable threat. Again, the facts of how they determined it, is unknown to us.
The question asked by the OP was whether this was legal, thus was my point of contention. As you state, we don't know the answers to your 3 questions, but would they make any difference to the legality of what the police did? They had a legal search warrant that included guns as the reason for the search. Along with the 2nd Amendment protecting our RKBA, the 4th protects us from unreasonable searches and seizures. We can't claim one is Gospel and the other is moot.
In many states, that is exactly the case. Having someone living in your home and using it a a legal address, does means possession and is trouble to the prohibited person.
My whole point is and has been in this thread, that we do not know enough of the real facts to make anything else but assumptions. As with many of these types of media releases, there has not been any follow up news that I can find to clear the fog. No where have I told folks to give up their guns without proper cause, only that it is silly to think one has a choice, if and when the cops show up at your door with a warrant looking for guns. I guess there is a choice, but for an individual citizen against several squads of police....... I would not. If it's an illegal search and seizure, I will trust the same constitution that protects my RKBA, to get them back for me.
We do NOT need to go there. And we can all do our own "safe keeping". 
