The government doesn't need to run them, they can just subpoena information when they need to do so. Moot even testified in the NFL bomb hoax thing. Users of 4chan, at least used to, delight in turning each other in, especially pedophiles. Here is a relevant section of wikipedia.
Doubtful any of them would need to be created directly by a government to still be a "honey trap"
Does it matter whether or not the government "runs" it if the outcome is the same either way?
Not to my knowledge. I do remember it's new owner was in hot water before for selling user data back on 2channel. So who knows.
Generally no maybe some indirect funding at best. GLP is a notable exception which iirc gets some direct gov't funding and direction through a think-tank nonprofit.
