How can one act correctly and appropriately towards his mother who treats him with daily harshness, directs insults and shouting at him, and prefers his siblings over him, which negatively affects his psyche and makes him feel distressed and angry?

The mother's right is great, and Allah the Almighty has coupled the right of parents with His own right. One must be dutiful to the mother and endure her annoying actions and be patient with her, for there is a reward in that. Her actions might stem from her love and concern for your welfare. Any flaw in her treatment can be rectified by speaking gently with the father. Being dutiful to her increases her love and affection. There is no sin in disliking her actions as long as it does not lead to disobedience. This dislike should be removed by remembering her kindness and her great right.