In the beginning of Dune, Paul is threatened with being killed if he fails the test of the Gom Jabbar. Jessica knew about it, but did Leto? If so, why did he allow it?
It seems unlikely that the Bene Gesserit breeding program would have remained a secret for so many generations if they had been in the habit of letting the nobility know about their tests and movements. As with any long term experiment involving humans the loss of secrecy would likely have been fatal to the breeding program or at the very least would have allowed recalcitrant Houses to use their gene lines for political leverage over the Sisterhood.
Granted it is a slightly abnormal situation as we are told that Jessica genuinely loves Leto and was willing to interfere with the breeding program for him. Nevertheless it seems unlikely that she would jeopardise her relationship with the Duke by admitting that she was allowing potentially fatal experimentation to be carried out on their only son by the Bene Gesserit.
It is never mentioned in the book itself so on the surface, no Leto I did not know about the Gom Jabbar test on Paul. It would be safe to say that given everything else going on, the move to Arrakis, the plots against him or even that the Reverend Mother was on Caladan, that Paul and Jessica kept this information from him. It is not even known how much information the ‘public’ knew about the Bene Gesserit and their quest for the Kwistz Haderach so the fact they were there to test Paul remains a mystery. We are in a universe of Dukes and Barons and Emperors it is, much like Star Wars, a very brutal world (such is Paul’s constant arms training and poison snooping) it would be surprising if no one of the Dukes staff knew about this visit let alone what it was for.
I doubt it. Jessica was still loyal to the BG, even if she was torn a bit due to her love for Leto. I doubt Leto was even aware that Paul was receiving Prana Bindu or any BG training of any kind, as the details of such are closely guarded secrets of the BG. Jessica never could completely let go of her loyalties to the Bene Gesserit. But before the events of Dune she was much more loyal than after.
It’s said in Children of Dune that the ability of the Bene Gesserit to control their bodies on a molecular level is kept secret out of fear for what would happen if the population heard about their ability to stop their aging. They were so afraid of this jealousy that they wouldn't use the ability, and would assassinate any member of the BG who did use it. All non-BG trained people seem to know is that the BG are "witches" and have a "weirding way", but no details as to what those actually are. The only specifics people outside of the BG seem to know about their abilities is Voice, and their abilities as truthsayers, and the former only seems to be known to nobility such as the Baron.