1. I believe that every person has a right for their opinion. I'd try to approach the individual and ask for why he has a problem with the servers or the team. I guess your case goes for a more stubborn person, so if he will just be irrational I'd actually ignore him as he can freely leave the server if he feels he has a problem with anything. ; Also insulting is not a rulebreak unless it goes down to racism and equally bad ways to insult groups of people or their ethnicity. In that case I'd silence that individual. He can then appeal.
2. As mentioned in 1 as for the insults, I would try the diplomatic approach first and try to resolve misunderstandings, however if it goes as far as to threaten me I'd probably inform the administration team just in awareness of a potential danger from someone that threatened me and could go as far as to widen it to the whole skial servers. Honestly I wouldnt know how to react, because up to that point nothing really happened. It could be an empty threat, yet like mentioned before, I'd make notice of him to the other administrators or post it in the administration section if such a section exists. If he would hack and DDOS the server, the case is easily ended with a permban because I am totally sure this will not be tolerated here. In that case:
sm_ban "#[U:1:12345]" 0 "DDosing servers"
3. Well first of all I'd take the case serious and try to deal with it as soon as I take notice. I will be honest, I have no jailbreak experience, but I assume a "freekill" means somebody is killed illegally (i was told some background of jailbreak). So if that happened, that obviously is rulebreaking, i just checked that up. Now back to the question:
I am called through Steam, I'd ask for further explanation of what happened, ask for infos. Hopefully there were screenshots taken that back his accusations up, because I wasn't there and cannot judge over what happened. Another possibility could be to check the chat logs, probably other people that were there have been complaining aswell through chat about that freekill (however I guess that would be a hard effort, yet mentionable as a possible solution). If the person has taken screenshots I'd take a look at them and judge depending of what I see. It must CLEARLY be visible that this has been a freekill. I can not think of many scenarios of how a screenshot could back that up, yet like mentioned before, probably there are other people complaining about it in chat in those screenshots.
If the proof is unaccurate or non-solid, I'd tell him that I need better proof than that or that if he is unhappy with my decision, he can file a report on this forum for another opinion (idk if that's an option). I would ask him to make sure to screenshot the situation or record a demo next time it happens, if necessary explain how to record one too.
4. If the guy is getting votebanned it needs 90% of yes votes and the whole server is whining? What kind of people would that be? Schizophrenics? (Obviously taking humour in this one :) )
Back to your question, I'd kind of react like stated above this sentence, I'd say that these people as a whole unit/server have passed that vote by themselves when they pressed yes. The other 10% are just unlucky and I cannot just change that vote because of a minority that thinks different, there'll be a reason the majority wanted the person gone.
Or you mean that they are whining cause he hacked, not that he was voted off: In that case if enough people support on server that this person has been hacking, I could look up hlxstats, get his SteamID and then put the temp ban into a permanent one. Hackers do not need to be part of our community. Hopefully people will be happy about that and the whining rests and joy returns.