Yes, they did. Eventually, Leia contact her good friend Pooja Naberrie to discuss her mother so they definitely knew about Padmé Amidala and the fact she was their biological mother.
Leia's Force abilities and her lack of awareness of her true identity also contributed to Vader's inability to sense their familial connection. Vader's denial of Anakin Skywalker's legacy and his fear of getting hurt again prevented him from realizing the truth about his children until later in the original trilogy.
In the Expanded Universe, yes, they finally find out after stumbling across recordings that Artoo had secretly made. Yes, in both. In Disney canon she knows about Padmé (and thinks quite a bit about her) in the book Bloodline by Claudia Gray.
It would appear that Mon Mothma, Padme's close friend and confidant, was also aware of the Skywalker secret, though she likely was unaware of Darth Vader's true identity. Luke and Leia's existence after the rise of the Empire remains one of the galaxy's most closely guarded secrets.
Palpatine tells Darth Vader that in his anger he killed her, placing the blame for Padmé's death on Vader. Vader denies it, saying he sensed Padmé was still alive, which the audience knows she was as Obi-Wan escaped on the ship with her.
Yes. In the book 'Tatooine Ghost', Leia learns of Anakin's life prior to his evolution into the Darth Vader personae. This revelation leads her to forgive him. She even named her third child in memory of her father.
Anakin Skywalker's birth mystery is solved - his mother, Shmi Skywalker, reveals that there was no father and she can't explain what happened. Qui-Gon Jinn believed that the Force itself created Anakin in Shmi's womb, making him the Chosen One who would bring balance to the Force.
Leia was not using the Force or was currently Force-sensitive during her confrontation with Vader. She began training as a Jedi years later. All these reasons prove my point: Vader didn't know Leia was his daughter, and felt no presence of the Force in her, as she was not currently strong with the Force.
He terrorized the galaxy at his masters side for over twenty years. Luke however did forgive his father, Anakin Skywalker. He knew he was manipulated by Sidious. That his father was a good person.
Luke and Leia never sensed that they were twins until Return of the Jedi despite both being sensitive to the Force. Anakin Skywalker's twin children are the original Star Wars trilogy's driving force, having been separated after Padme's death.
Darth Vader acts like he doesn't even know C-3PO or R2D2. They were with him since he was a child. He did everything he could to forget his past. Also, his plating had been replaced many times, and protocol droids all look very similar.
Leia's ordeal. In the rough draft of Star Wars: Episode IV A New Hope, Darth Vader tortures Princess Leia with powerful electroshocks. In the incarnation of the second draft of A New Hope, Leia is replaced by Deak Starkiller, the brother of Luke Starkiller (who would become Luke Skywalker in later drafts).
Anakin does not have even the slightest bit of remorse. By the end, he feels nothing but anger. Although he feels regret for killing Padme, he still helps the evil Emperor. The film ends with Anakins standing side by side with the man on whose behalf he has committed all his crimes.
Leia and Han had three children, and they named their youngest son Anakin after Anakin Skywalker, Leia's biological father. Since Luke was the twin to spend more time with Obi-Wan in the old canon, it made sense that he, not Leia, was the one to honor Obi-Wan when naming his child.
The canonical comic Star Wars: Darth Vader #7 confirms the younglings' deaths caused Anakin grief, self-loathing, hatred, and pain, fueling his descent even further into the Dark Side.
The subtitles and script show some of Padme's final words to Obi Wan as “There is still good in him.” Every time I watch this I swear she says “Obi Wan... there's quite a name.” I have always thought of it as her recognizing him for everything he has done over the trilogy.
Honestly, I just couldn't figure out who she was or why she was even in the funeral procession. This also means that for the longest time I forgot I could just look up the credits of the film. Long story short, she is Padmé's niece, Pooja Naberrie, and was portrayed by Hayley Mooy.
Anakin's vision did come true, as Padmé, heartbroken by her husband's downfall, loses her will to live and succumbs while giving birth to twins. Before her death, Padmé still believed there was still good left in Anakin and that one day he could be saved with the help of their children.
