Can an ongoing charity be performed on behalf of the deceased, and what are its forms, such as placing Qurans in the mosque? Does a mother's prayer for the deceased atone for his sins? Is it permissible to make up missed prayers for the deceased due to his illness?

"Ongoing charity" (صدقة جارية) is that whose principal remains intact while its fruits are benefited from. It is valid from a Muslim during their lifetime, such as Umar's endowment for his share in Khaybar, or after their death. Placing copies of the Qur'an in a mosque with the intention of charity is considered ongoing charity. Supplication for the deceased benefits them, especially if it comes from their mother. As for performing prayers on behalf of the deceased, it is not legislated; rather, supplication for them and giving charity on their behalf are more preferable.