Hassan tells Amir about the dream they had as kids in order to reconnect with him and possibly bridge the gap that has formed between them. It is a way for Hassan to show his loyalty and affection towards Amir, hoping to rebuild their friendship and trust.
The dream that both Hassan and Amire share is the one that they save the people by defeating the sea monster in the lake.
Amir is selfish and Hassan is humble and loyal to Amir. Hassan is a Hazara, Amir is a pashtun. Amir is granted to right to get an education, Hassan is not.
Hassan doesn't tell the truth about Amir to protect their relationship and honor. He values their friendship and doesn't want to break the trust between them, even if it means taking the blame for something he didn't do. Hassan's loyalty and selflessness are stronger than his desire to seek justice for himself.
Hassan uses a slingshot to defend himself and Amir against Assef when he tries to attack them. Hassan ultimately sacrifices himself by getting severely beaten rather than giving in to Assef's threats.
Hassan chooses not to tell on Amir because of their strong bond of friendship and loyalty. He values their relationship above all else and is willing to protect Amir, even at the cost of his own reputation or safety. Additionally, Hassan may feel grateful for the kindness and care Amir has shown him in the past, leading him to forgive any wrongdoings.
Actually, they are half brothers. Amir had found out that when Rahim Khan had hinted that Ali was not Hassan's father but Baba was (which explains why Baba had a closer relationship to Hassan and Amir).
Amir frames Hassan for stealing money and a watch, knowing that Hassan will take the blame to protect him. Amir stands by and does nothing to stop it, thus indirectly getting both Ali and Hassan in trouble.
Amir threw pomegranates at Hassan out of frustration and jealousy stemming from their complex relationship and his own insecurities about his father's affection for Hassan. This act was an expression of Amir's conflicting emotions and desire to hurt Hassan in the heat of the moment.
Ali tells Amir that Hassan mostly runs errands and does chores for him now, as he is getting older and can no longer do physical work like he used to.
Amir and Hassan, the sultans of Kabul
On pages 73-74 of The Kite Runner, the two memories and dreams presented underscore the guilt and longing that protagonist Amir feels towards his childhood friend Hassan. The memory of Amir throwing pomegranates at Hassan and then his recurring dream of being pursued by a faceless monster reflect Amir's internal conflict and remorse over betraying Hassan, highlighting the complexities of their relationship and Amir's desire for redemption.
Hassan and Ali give Amir a leather-bound notebook for his birthday.
Hassan's first word is Amir. Amir does not reveal his name till this point, which shows his underlying guilt because Hassan loves him so much and Amir's betrayal.