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.
It wasn't until Bloodline that she began to truly understand Anakin Skywalker, in a situation where she too felt tempted. "She'd always wondered what had led her father to turn to the dark side," Leia reflected, "to become Darth Vader. She'd imagined it came from ambition, greed, or some other venal weakness.
Although we never see Leia fully reconcile with the fact that Darth Vader is her father, it's worth noting she did actually forgive and accept the former Sith Lord in the non-canon Legends novel "Tatooine Ghost," by Troy Denning.
In Star Wars Legends Leia was visited by the force ghost of her father Anakin Skywalker, the man who used to be Darth Vader. It wasn't a pleasant meeting as Leia was unable to forgive him for the things he'd done no matter how hard he begged, and it was the only time she saw him.
Luke gave him unconditional love, but that's not forgiveness. Anakin, as Darth Vader, committed and was complicit in a whole bunch of genocides and other crimes. The films never suggested that these are forgiveable. Furthermore, Anakin's redemption arc comes in that form of concrete action.
In Episode III, just before they are to set off on their respective missions, Obi-Wan and Anakin share a farewell. Anakin admits to Obi-Wan that he has been arrogant and unappreciative of his training, and apologizes to his master.
Yes, when Darth Vader turned back to Anakin Skywalker and sacrifice himself to save his son, he dies and Obi-Wan uses the force to connect with Anakin's spirit. In that time Anakin apologizes to Obi-Wan, desperately asking him to forgive him for everything he has done to him and everybody over the years.
Despite participating in Leia's interrogation, the Sith Lord doesn't uncover his familial connection with the rebellious royal. In contrast, Vader sensed his connection to Luke Skywalker almost immediately.
In the darkest hour, Padmé Amidala became a Force Ghost. Padmé, knowing that her childern should be kept in hiding until the time was right, kept this new found ability a secret. As he heard of the news, the spirit of Anakin Skywalker in shock of what he did on Mustafar, disappeared forever into the Force.
Luke begins to tell Leia about how Anakin Skywalker was redeemed and what his last words were (“tell your sister you were right”), but Leia doesn't want to hear it. Her history with Vader, being personally tortured by him, makes it all the harder for her to hear or accept any of this.
It is unclear whether Yoda would be more or less forgiving than Obi-Wan but one thing is certain - the former Grand Master would be ready to treat Anakin as an ally, following the events on the second Death Star. In Star Wars: Revenge of the Sith (2005), Yoda never really lost faith in Anakin in the way Obi-Wan did.
Why didn't Darth Vader recognise Leia as his daughter in Star Wars: Episode IV? He didn't recognize her because he didn't know his daughter was alive, and he couldn't tell it from reading her mind because she wasn't aware of it herself.
He only senses Luke when he actually using the Force, Leia is not actively aware or using the Force and therefore he does not sense her. He is also not looking for a daughter either until Luke spills the psychic beans on her later on.
Yoda didn't specifically know how this would go down, or to what degree Anakin would cause it to happen, but he was playing a much longer game by Revenge of the Sith, and that's at least, in part, explained by Yoda's Force connection with Qui-Gon's spirit.
Similarly, her rebellious, rule-breaking and trouble-making nature resembled Anakin more than Padmé. However, both parents were adventurous and stubborn people. Thus far, her leadership skills are the one thing missing from Leia's portrayal in Obi-Wan Kenobi. It's well justified since Leia is still a child.
Darth Vader was aware that Padmé Amidala died. That's because Palpatine told him what happened to Padmé after he saved Anakin on Mustafar. In fact, there was a big funeral held for Padmé on Naboo. But Vader didn't know that Padmé died of a broken heart as he thought she died by his hand.
However, when Palpatine destroyed the Republic, Panaka sided with him. Though he never acted against Padme, his support of Palpatine included the support of a stronger military presence, which was a betrayal of her ideals.
In the post-2014 canon universe, the story that reveals Darth Vader discovering Luke Skywalker's identity (and thus the fact that the Death Star's destroyer is his son) is issue 6 of 2015's Star Wars comics by Marvel, which takes place sometime shortly after A New Hope.
We go for the more simple answer, further considering the details of the question itself. If Anakin didn't go into that chamber as the Jedi confronted Palpatine, then Mace Windu would have defeated Palpatine, killing him. If you remember, Mace had him. He was ready to kill him.
He does. Right at the end. With Luke crying over him and Vader sensing his anguish and pain he had inflicted on all those who had cared for him.
If you are asking about Vader post-Mustafar, then no, she doesn't. Padmé doesn't seem to know Obi-wan cut off his limbs and left him to burn alive, Obi-wan didn't believe Anakin to be alive and Anakin's surgeries were still happening as Padmé died so there was no way for her to know.