Obi-Wan apologizes for what he did to Anakin, but Vader assures him that he killed Anakin, not Obi-Wan. He then goes on to proclaim that he will destroy Obi-Wan in the same way.
“You didn't kill Anakin Skywalker. I did.” This allows Obi-Wan to finally let go of all his guilt over Anakin's transformation into Darth Vader. It allows Obi-Wan to resume some kind of life, but it's a tragic reminder of how Anakin's fate is captured behind the black mask.
Of course, Obi-Wan didn't finish Anakin off ten years ago on Mustafar because he didn't want to be the one to strike the killing blow on a man that he loved like a brother.
Anakin Skywalker ultimately ended up hating Obi-Wan Kenobi for many misguided reasons, but Darth Vader's subconscious response to the Jedi's final declaration of love in the Star Wars prequels shows the real reason he could never forgive him.
“I am not your failure, Obi-Wan,” Vader says. “You didn't kill Anakin Skywalker. I did.” With that closure, Obi-Wan is able to walk away from the encounter, acknowledging his old friend as “Darth” and leaving his attachment to Anakin behind.
Alsoa reason he tried to forget and not think about his past was because of sidious. vader knew that palpatine could sense what he was thinking about and sidious would punish vader for reflecting on his time as anakin because sidious would see that as a weakness in vader.
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.
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.
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.
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.
He felt that Kenobi and the other Jedi Masters were simply jealous of him and his power. Only during their duel on Mustafar, Anakin finally allowed his feelings to take control of him. He finally let out all of his frustration, and that was why he had the guts to tell Obi-Wan that he hated him.
But, on top of all of that, the biggest reason Obi-Wan doesn't know Anakin is alive is that the last time he saw Anakin Skywalker in Revenge of the Sith, Ben literally left him for dead. Ten years later, in Episode 2 of Obi-Wan Kenobi, he learns Anakin is alive. This is the first time it happens.
Darth Vader only learned he had a daughter in Return of the Jedi - but there's no evidence he realized she was Leia before his death and redemption. Darth Vader may have never known Princess Leia was his daughter in Star Wars and never cultivated a relationship with her, even when he was a Force ghost.
However, Leia's strong-mindedness during Vader's probe denied him the opportunity to discover she was his daughter. That serves to justify Vader's lack of awareness of his connection to Leia, although, perhaps, not as satisfyingly as if Lucas had planned for the relationship from the beginning.
The reason Darth Vader could sense Obi-Wan and Luke was because of how they actively used the Force several times throughout the Original Trilogy. Vader never sensed Leia because she didn't the Force; Leia did not even know she had Force powers to begin with.
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.
David Porter provides some great philosophical context to why Obi-Wan would not regret leaving Anakin alive, as it was obviously “the will of the Force” that Obi-Wan be unable to finish him off, and “the will of the Force” that Anakin live on to kill Sidious himself in the future.
He explained to Luke, “I took it upon myself to train him as a Jedi. I thought I could instruct him just as well as Yoda. I was wrong.” His guilt was apparent in Revenge of the Sith, when he explicitly told Anakin in the heat of their duel on Mustafar, “I have failed you, Anakin.
He had a softish spot for a handful of individuals and unlike many other Sith, Vader didn't generally kill innocent people for no reason - but he always hated the Jedi and he had no regrets about what happened to them. The best proof of this is how he throws himself into becoming the Empire's primary Jedi hunter.
Giving in to his feelings, Anakin cuts off Mace Windu's hand, giving Palpatine the opportunity to kill him. Anakin regrets this almost instantly, but the thought of Padme was stronger than his remorse This is the moment Anakin officially became Darth Vader and led to serious issues in the Skywalker Saga.
Did Anakin regret choking Padme? Oh yeah. He pushed the memories out of his mind for a couple of decades. But after he faced his son, Luke, in Empire Strikes Back, it was all he could think about.
Yes, his entire character since then was a self loathing tragic monster who has little reason to care about anything. Betraying Mace Windu instantly made him regret his act of impulse, Padme's death and rejection of his actions later on made it stick.