Is it prescribed to fast some nights and to pray and supplicate for someone who has done me a favor, and to do that for my living mother who raised me? Is it permissible to express gratitude by reciting the Quran as a way of returning the favor? Do these acts require a specific formula of dedication, or is the intention of the heart sufficient?

The Hadith encourages rewarding those who do you a favor. If you find nothing material to give, then supplicate for them. If you say, "May Allah reward you with good (Jazakallahu Khairan)," then you have expressed the utmost praise. Supplication during prayer, especially in prostration, is more likely to be answered. Supplicating for the living and the dead is permissible, and a Muslim's supplication for his brother in his absence is answered, and he receives the like of it. There is no harm in supplicating for others while omitting supplication for oneself. As for dedicating the reward of reciting the Quran, it is a matter of dispute; some scholars permitted it, while others prohibited it. It is best to confine oneself to what is explicitly mentioned in the texts, such as supplication, seeking forgiveness, and charity. If the one dedicating the reward intends for the reward of the deed to reach a specific person, that is sufficient.