What is the ruling regarding my mother's recitation of the Qur'an, especially Al-Fatiha, given her difficulty in distinguishing some letters, and is her prayer valid?

Your mother should strive to learn the correct pronunciation and perform it to the best of her ability. What she is unable to master will be excused, because she has feared Allah as much as she could. Scholars have made allowances for letters with similar points of articulation, especially for the common people, due to the difficulty of being precise.