Your idea of the limitations of our right to free speech seems to depend on a referee; otherwise, it seems that you are conflating the idea of 'rights' with 'legality'.
> I have no right to say "Hey, so-and-so is a cheat, don't do business with them," however strongly I believe it, when they're not.
Who gets to say they are or are not a cheat? Spreading false rumours is reprehensible, but within your right to speak freely: just like the so-called cheat has the right to sue the rumour-maker for defamation.
> I have no right to say "You should rob a bank, and I'll teach you how."
It's illegal to encourage/enable illegal actions, but you have the right to do illegal things - just like you have the right to bear the consequences of your illegal actions.
> I have no right to say "This herb will cure your disease" merely because I believe it, if nobody has convinced the government to also believe it.
What if the herb (or cutting-edge treatment developed in some other country) works beautifully in another country, but the FDA in the US doesn't recognize it yet? You've set up the US government as the referee who gets to decide when we are allowed to have our right to free speech.
> I don't even have the right to repeat your comment, except in specific ways and circumstances.
You have the right to repeat the comment however you like - but you also have the right to be sued under copyright and defamation laws.
> fundamental rights must be preserved even when used distastefully
They are also, as recognized in the US declaration of independence, unalienable: the government has no say in whether we get to have them.
> not at all clear to me that propagating dumb conspiracies in a way that leads to violence is within the line...since it's not the conspiracy itself that makes it questionable here
As you point out; someone theorizes about a Bad Person, someone reads that conspiracy stuff online, and decides to 'do something' about it, and tries to shoot that Bad Person. In the US Code, inciting to commit a crime looks like this: "whoever...solicits, commands, induces, or otherwise endeavors to persuade such other person to engage in such conduct".
So what exactly do you mean by "in a way that leads to violence"? Because what the US code describes is vastly different from "theorizing about the existence of a conspiracy", however 'out there' the conspiracy is. And in at least the example of the pizzagate shooter, he persuaded himself of his actions - the conspiracy just made him think that a Very Bad Thing was happening there. And there is nothing wrong with theorizing about the existence of a conspiracy of people doing Bad Things.