Is the trouble of the story because characters didn't drew the line between friendship and master/servant relationship?

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The characters of Kite Runner have somewhat a very complicated relationship between them. Baba and Ali have a master and servant relationship mixed with deep friendship that traces back to Baba's family history. Amir and Hassan have a very similar relationship to Baba and Ali. They are half brothers (they had the same wet nurse), friends, while maintaining a master/servant relationship.
Because these characters have developed kinship between each other, it interferes in the simplicity of some occasions. If Baba did not have affection for Ali and Hassan, maybe things would have been easier. Baba would not praise Hassan for his excellence and would not do nice things to Hassan. Amir’s jealousy of Hassan drives the entire plot. It is witnessed all the things Amir is willing to do to conquer the attention of Baba. When asked if he would even get new servants, Baba gets angry at Amir and says he will never leave Ali because Ali is family. This event destroys the brief happy father/son relationship Amir and Baba had.
Amir would treat Hassan poorly because Hassan is a Hazara and his servant. He would find reason in their master/servant relationship to justify his wrong doings but feel guilty afterwards for having mistreated a brother.
Their brotherhood made Hassan defend Amir from Assef, angering Assef. Later in the story… (we all know what happened). Hassan’s love to Amir brings big pain to him. In the attempt of defending his good brother, he angers a sociopath; and due to the burden of having to bring back his best friend’s prize kite, he receives a scar from a barbaric act from Assef. This is all could have been avoided.
Amir spends a great deal of effort feeling guilty for not treating Hassan right. Deep in his heart he believe in his brotherhood with Hassan. However, his love for Hassan is ruined because of their place and time in history.

I totally agree with you in that Amir and Hassan's conflict comes from their complicated relationship. If they were friends only, they would have been better off: Hassan might still protect Amir out of brotherhood, but at the same time Amir might protect Hassan as well, just maybe, but much more possible than if Hassan is Amir's servant. On the other hand, if Hassan were just the servant of Amir, things might have been bette too since Amir no longer struggles.

What I want to ask is, though, how can Baba and Ali live together without trouble with the same kind of cplicated relationship? It is apparent that Baba treats Ali both like a servant (Ali take care of the household) and a friend (Baba got pissed off when Amir asked whether they could change a servant). Is it because of age, as Amir is still young and naive, or is it because Baba somehow overcome the compliacted relationship?

In my own opinion, I think it is impossible to clearly divide the line between master/servant and friendship, also there is something different between Baba/Ali’s relationship and Amir/Hassan’s.
When the master/servant relationship exist, it is determined that someone in the relationship will be more dominate and superior. This concept will root in both mind and alter their behavior to fit’s this relationship (In order to meet social expectation.)
For Baba and Ali, since Ali’s parents are both died from accident, Baba may feel sympathy toward Ali and thus, treat Ali better due to pity. For Amir/Hassan’s relationship, this bond didn’t build upon sympathy, Amir may feel like Hassan didn’t deserve this kind of favor from Baba.

The root of this issue is probably racism. All Hazaras occurred in the story are servants. When Amir is searching for Hassan, the stranger he met said the Hazara (Hassan) should swept his owner’s feet with his eyelash. So imagine if Ali is not a Hazara, then when Baba’s father adopted him, he would be treated exactly as a second son, rather than a son-like servant. The relationship between Ali and Baba would then be precise, so that the confusion of friendship and master/servant relationship of their next generation wouldn’t exist.