Do the other houses learn of the betrayal of the emperor and Baron? Does mohiam? If so, was he asked about this?
No, but we don't really see much of the other houses. It would be kept secret as if the Emperor can try and destroy House Attredies, they could do it to any House, which would lead them all to band together against the Emperor. The Bene Gesserit know about it (they know about everything), and tacitly approve of it as they can still use the bloodlines, but above all they don't want the flow of Spice to stop, and they want to keep the status quo (where they have a lot of power), at least until the get their own Kwizatz Haderach.
I understand that BG didn't want a male KH, why did mohiam want him spared with Jessica?
They do want a male Kwizatz Haderach. Only a male can become a Kwizatz Haderach, but the Bene Gesserit want one completely under their control. The Bene Gesserit have doubts in the beginning that Paul is the Kwizatz Haderach (he may be one of the many failed Kwizatz Haderachs), as he is a generation early, and they hope that they can still access the bloodlines to breed their own, or maybe even control Paul as their own superbeing.
Does Paul have any dialogue with Baron?
I don't think they ever do talk face to face. I can't remember if they have any dialogue after Paul takes the water of life and accesses the genetic memories and personalities of his ancestors, as Vladimir Harkonnen is Paul's maternal grandfather.
Alia certainly has a lot of conversations with Baron Harkonnen's personality in Book 3
How did they know liet 'betrayed' the emperor? In the movie, was she about to summon a worm to ride before being stabbed?
Keynes goes native. In the book, he's Chani's father, and walks a fine line between being part of a Fremen tribe, and doing his job for the Emperor (while secretly working in the Fremen's best interests). IIRC, the Harkonnen simply decide he can't be trusted due to his split loyalites, and how he won't give them the answers they want regarding the fate of Paul and Jessica, and the true facts about the Fremen. Keynes counts on his being a direct apointee of the Empire to protect him, but the Harkonnen arrange for an "accident" that means his body is lost to the desert and worms.
How did the fremen come to know of the prophecy? Is that what the BG instilled throughout the years?
You've got to understand how humanity has spread across the stars in the first place. There have been many diaspora in the the thousands of years before the Butlerian Jihad (that banned "thinking machines"). This allowed humanity to spread to the stars before the discovery of Spice and the founding of the Spacing Guild. Before the Spacing Guild was formed and the Butlerian Jihad was enacted, people use machines to travel to stars, and now they use the Spice mutated Navigators of the Spacing Guild.
In those first and many diasporas, the Bene Gesserit planted the seed of their religion, many versions of their stories and beliefs. These belief systems were kept whole by Bene Gesserit Mothers being embedded in the native tribes and peoples (rather like missionaries or the priest of a diocese). This allows Bene Gesserit to turn up and portray themselves as mythical or messianic figures in order to protect themselves and control the people they meet. It's a work of tens of thousands of years. This is what happens when Jessica and Paul turn up in the desert after escaping the Harkonnen attack. By saying the right things and acting the right way, the Fremen understand instantly that Jessica is Bene Gesserit with powers and skills (and soon a Reverend Mother), and that Paul is the messiah they've been praying for generations would arrive to lead them to paradise.
It's as if someone turned up turning water into wine, feeding thousands with a few fish and loaves, making the crippled walk and the blind see. You'd understand culturally who he is, especially if you're a very religious people living in a very harsh environment, and seeing the embodiment of your god Shi'Halud on a regular basis. You'd know he's a messiah, and if you believe that, then you'd believe it's your god-given destiny to follow his Jihad against the evil Harkonnen and Emperor to rule the galaxy and bring a paradise to your people that have endured so much to get to the point where their messiah arrives to bring them to a heaven of their own on Arrakis.
Of course, this isn't just a confidence trick on behalf of Jessica and Paul using the safety nets seeded by the Bene Gesserit thousands of years before. Paul actually is the Fremen/Bene Gesserit messiah, forced by breeding and training to the point where the spice-laden environment and his decision to take the water of life to aid his survival is the final step to making him the Kwizatz Haderach.
