I don't think that Baba gains his redemption in the novel. He tries to by building the orphanage, by treating Hassan extremely well while he and Ali still live with Baba and Amir, and by finally accepting Amir's writing at the end of his life. However, he never tells the truth to Amir or Hassan about his fathering Hassan. By taking that information to his grave, he keeps Amir from perhaps being able to save Hassan and Sohrab; for if Amir knew that Hassan was his half brother even just a couple of years before Rahim Khan breaks the news to him, he might have tried to make things right with Hassan. Baba never, as far as the author implies, tries to rescue Hassan and Ali from Afghanistan even though he obviously followed political events in the country and must have known that the situation there was grave.
In a sense, Amir's own redemption is partially for Baba, but then one must ask if one person can redeem another.
There are a couple of errors in the third post. Baba does not ask Rahim Khan to go get Hassan and bring him back to the Kabul house; this is something that Rahim does on his own. Also Rahim Khan contacts Amir on his own, not because of Baba's dying request; again, he does this on his own as part of his redemption. See Chapter 16 which is told from Rahim Khan's point of view.
Baba gains his redemption in the novel in several ways. However, the sad part is that these things take place only after his death. First, I would say that the truth comes out about Amir and Hasan later in life. And he is not judged by Amir, but Amir understands. Second, what Baba wants is for Amir to become a man of integrity and strength and this is exactly what happens as he faces his past and embraces an honorable course of life. Finally, Hasan's seed through his son lives on. This is to say that Baba's grandson lives on through Amir. These points should get you started.
Baba is the true father of Hassan in the story "The Kite Runner." He does not tell the child nor does he tell his son Amir. Instead the boy lives with his father, Baba's servant whose wife Baba had laid with, and also works as a servant for the family. Hassan and his father leave following Hassan's rape and Amir's setting him up by hiding his watch in Hassan's bed.
The first act of redemention is that Baba's friend seeks out the child Hassan. He finds Hassan who now is married and has a son. He has him brought him back to his home his homeland into work. The second act of redemption occurs when Amir's father, Baba, friend and confident contact Amir following his death. He lets him know that Hassan's son is in great danger.
The redemption occurs when Amir steps into courage and rescues the son of his brother Hassan. Through his acts of courage it is Amir who has helped his father's redemption. He has saved the son of Hassan and will raise the boy.