Important Questions about Classic Movies

(1) Star Wars Trilogy
Background: So after seeing the prequel trilogy, we know that Vader, the Emperor, and a bunch of the other bad guys knew that Vader impregnated Padme. The surviving Jedi (Obi-Wan, Yoda) and the Organa family were there for the birth of Luke and Leia. We're not sure whether Vader and the Emperor knew that (a) Padme actually had the children and (b) Padme had twins. In any event, the Jedi + Organa brain trust decide to put Luke with his aunt and uncle on Tatooine to get him far away from the Emperor and Vader and put Kenobi there to keep watch. Then they decide to put Leia with the Organas, a powerful and influential political family, on Alderaan, but no Jedi to keep watch. And then Yoda goes off to a random spot to do whatever.
Questions:
(a) Why didn't Yoda go with Leia as her watcher/protector similar to Kenobi with Luke?
(b) When and how did Vader find out about Luke being his son? Episode IV in the Death Star trench? "The force is strong with this one, must be my son?" Episode V? If so, when?
(c) If Vader knew about Luke, how did he not know about Leia? He was in very close proximity to her in Episode IV on the Death Star?
I'll have more of these, mostly centered around Star Wars.
Background: So after seeing the prequel trilogy, we know that Vader, the Emperor, and a bunch of the other bad guys knew that Vader impregnated Padme. The surviving Jedi (Obi-Wan, Yoda) and the Organa family were there for the birth of Luke and Leia. We're not sure whether Vader and the Emperor knew that (a) Padme actually had the children and (b) Padme had twins. In any event, the Jedi + Organa brain trust decide to put Luke with his aunt and uncle on Tatooine to get him far away from the Emperor and Vader and put Kenobi there to keep watch. Then they decide to put Leia with the Organas, a powerful and influential political family, on Alderaan, but no Jedi to keep watch. And then Yoda goes off to a random spot to do whatever.
Questions:
(a) Why didn't Yoda go with Leia as her watcher/protector similar to Kenobi with Luke?
(b) When and how did Vader find out about Luke being his son? Episode IV in the Death Star trench? "The force is strong with this one, must be my son?" Episode V? If so, when?
(c) If Vader knew about Luke, how did he not know about Leia? He was in very close proximity to her in Episode IV on the Death Star?
I'll have more of these, mostly centered around Star Wars.