Is it permissible to supplicate that Allah grants us the understanding of the prophets and the memorization of the messengers, or is that among the exclusive characteristics of prophethood and messengership?

It is permissible for a person to ask God to grant him understanding and memorization like the understanding and memorization of the prophets. It is also permissible to supplicate during study and exams with the mentioned supplications, even if their specific timing is not mentioned in the Sunnah. Ibn al-Qayyim encouraged supplicating with the Hadith: "O Allah, Lord of Jibra'il, Mika'il, and Israfil..." when facing difficult issues. He also narrated various supplications from the Salaf when giving fatwas, such as "O Teacher of Ibrahim, teach me," "Glory be to You, we have no knowledge except what You have taught us," and "My Lord, expand for me my breast." All of this depends on sincere intention and honest devotion to God.