What is the ruling of the religion, and is expiation obligatory for one who had intercourse with his wife 13 days after her miscarriage, based on a fatwa that permits purification, prayer, and intercourse, knowing that on the third occasion, blood came down during intercourse?

The blood your wife saw after the miscarriage is irregular bleeding (istihada), unless a human form was discernable in the fetus, the minimum period for which is 81 days. It is permissible to have intercourse with a woman experiencing istihada. This is the view of the majority of scholars and is the most preponderant opinion. There is no blame upon you for what you did by having intercourse with your wife, whether she was experiencing istihada bleeding or not.