What did Jessica know?
Jessica was aware of both the need and reason for her to bear a daughter as well as the Bene Gesserit long-term goal of producing the Kwisatz Haderach from the Atreides bloodline.
“So I had a son!” Jessica flared. And she knew she was being goaded into this anger deliberately.
“You were told to bear only daughters to the Atreides.”
“It meant so much to him,” Jessica pleaded.
“And you in your pride thought you could produce the Kwisatz Haderach!”
Jessica was kept unaware that she was the daughter of the Baron Harkonnen, presumably so that she wouldn't have to lie to the Duke about her heritage and to forestall any objections she might have to her daughter being wed to so close a relative as the Baron's nephew.
He said: “But my mother tells me many Bene Gesserit of the schools don’t know their ancestry.”
“The genetic lines are always in our records,” she said. “Your mother knows that either she’s of Bene Gesserit descent or her stock was acceptable in itself.”
“Then why couldn’t she know who her parents are?”
“Some do . . . Many don’t. We might, for example, have wanted to breed her to a close relative to set up a dominant in some genetic trait. We have many reasons.”
What did the Duke and the other noble houses know?
Given the depths of the Duke's antipathy toward the Harkonnen, it's unlikely that he was aware or would have approved of the most visible part of the short-term plan (to marry an Atreides daughter to the Harkonnen heir) although he was a supremely pragmatic man and could, probably have been talked into it.
“No.” She shrugged. “There’s good political reason–as long as my Duke
remains unmarried some of the Great Houses can still hope for
alliance. And . . . ” She sighed. “
He, and his fellow nobles, was obviously entirely unaware of the very long-term (and very very secret) BG plan to breed a superhuman capable of transcending time and ruling the galaxy.
Because the Bene Gesserit operated for centuries behind the blind of a
semi-mystic school while carrying on their selective breeding program
among humans, we tend to award them with more status than they appear
to deserve. Analysis of their “trial of fact” on the Arrakis Affair
betrays the school’s profound ignorance of its own role