Trolling and Repeated Trolling:
While we can all agree that trolling is a persistent and annoying phenomenon that needs to be managed, we likely wouldn't agree on one clear definition for what it actually is. This is problematic, as someone on the receiving end of moderator intervention might not feel like they did anything wrong. For this reason, my initial approach would consist of three things: identifying whether or not trolling is actually taking place, determining what action needs to be done, and communicating with the perpetrator.
Identifying whether trolling is taking place or not needs to be done via context, in my opinion. Is someone intentionally derailing the conversation? Is someone attacking someone, or bullying someone? Is someone trying to drown someone out? Is it a tasteless joke with no intention to troll? Is the opposing party in on it, and is it in the context of fun? Most importantly, is there a real need to intervene, and what degree of intervention is needed? A private word, or a public one? Deleting a post? Temporary time out? In my opinion, the softer the intervention the better, but it will certainly have to depend on the case. Once this is determined, communicating with the user before any action is taken is preferable. When things are simply too frantic for that, clear communication as soon as possible after moderation is a must, so as to avoid escalation.
Repeated or persistent trolling is another issue. If the trolling is done in defiance of a moderator's decision and despite clear and open communication, then a slightly heavier hand might be needed. Bel 3arabe el mshabra7, iza el wa7ad 3am ya3mol nkeyet for the sake of it, ma lezem ykoun fi ghenej (domn el 7doud el manti2iyye, akid). If the persistent trolling is directed towards other users, then I'd rather make sure the behavior is not being instigated by someone else before taking any action. In any case, communication needs to be clear so that everyone understands where they stand.
Fake News:
Deleting news is a slippery slope. A lot of the time, the truth of the matter doesn't appear until later, or until much later. Personally, I'd rather see unreliable news items and sources being labeled with an inoffensive tag (if technically possible), rather than being taken down. Fake news that is proven fake should be labeled as well, but not deleted. I have many more ideas regarding the matter, but it's something I'd rather discuss first (I'm not entirely sure what is possible and what isn't). Some football subreddits have tier lists for reporters, for example. We could also set up fact-checking threads, but this would require user engagement.