What are the principles for proposing to the sister of a friend, especially when the friend is her guardian and his anger is feared?

There are no specific Sharia (definition)-based principles for engagement beyond the manners and customs that align with Sharia. If you wish to propose to your friend's sister and fear his anger, the mere fact that he loves his sister is not an impediment. You can consult him about who among women he would advise you to marry, or you can subtly express your desire for marriage or kinship to strengthen the bond between you, and observe his reaction. If you fear his reaction, another person can broach the subject with him. Supplication is important in fulfilling one's needs.