Climate change isn't any kind of state dogma. It's a scientific fact. Cursitor Doom chooses to believe one more right-wing conspiracy theory - his demented dogma - but the people who lie about the fact are putting other people's live in danger, and really should know better. Executing them won't solve any kind of problem, and I wouldn't want it to happen, but people like him should be aware of the risk they are running.
Probably not. Their addiction to believing dangerous nonsense is more your kind of problem. At least you don't have enough power to act act out your delusions.
Flyguy confuses logical deductions based on his idiotic posts with libellous and misleading statements likely to lead people to treat him with ridicule or contempt.
The problem with that is that the ridicule and contempt is directed at the foolish and misleading statements that he has posted, and he would end up being shown to be trying to sue himself for the damage his own posts had done to his reputation.
Perhaps Flyguy should consider getting legal advice from somebody slightly more intelligent than he is, which does seem to include at least 95% of the population.
Sydney libel lawyers are an expensive bunch. Your chance of getting one that would work for you on a contingency fee basis would be slight to non-existent even if if you weren't a total idiot.