I’m a hard no on both questions.
First, I’m very certain that Varys is exactly who he and Illyrio says he is, a former mummer who turned himself into one of the most influential people in Westeros and Planetos.
Second, not only do I not think that happened, but I think the very idea would be an abomination for the Faceless Men. They’re not psychopaths, they’re a holy order.