Are they in control of the world?
No.
Like a lot of conspiracy theorists say they do.
Conspiracy theorists say lots of things.
Do they even exist?
Yes, there are groups that refer to themselves as Illuminati.
http://en.wikipedia.org/wiki/Illuminati#Modern_Illuminati Is the evidence for their existence based upon mounds and mounds of fallacies and misinterpretations?
In reference to the Illuminati of popular conspiracy theories, yes.