When does the last third of the night begin, and does the questioner's prayer 45 minutes before the Fajr adhan fall within it? And is it permissible to accept money in exchange for memorizing the Qur'an, or is it better to donate it as charity?

The method for knowing the last third of the night has been previously explained. Your prayer, if performed before dawn, falls within the last third of the night. There is no harm in benefiting from the amount paid by the Quran memorization center for memorizing a part of the Quran, provided that your intention is solely memorization and not coveting the money. You may also donate it if you wish.