Does the discharge that exits the anal orifice, which may be accompanied by a brown color and an odor, and which wets the inner garment, affect the validity of the prayer, and what is the ruling on past prayers in this case?

The area must be washed when performing istinja' (washing after defecation or urination) until it returns to its natural roughness, and a strong presumption is sufficient for achieving thorough cleansing. As for istijmar (cleansing with stones or similar dry materials), thorough cleansing is required until all traces are removed, except for what cannot be removed except with water. If a person sweats afterwards, it does not matter, and no attention should be paid to obsessive thoughts (waswas); for the area is considered pure as long as the istinja' has been performed in the prescribed manner. Al-Muwaffaq mentioned in Al-Mughni that Ahmad ibn al-Husayn asked Abu Abdillah about a man who urinates, performs istibra' (ensuring no drops remain), istijmar, and then sweats in his trousers, to which he replied: "If he performs istijmar three times, then there is no harm." Another person asked him about performing istinja' from feces and if water touches another area, Ahmad replied: "It has been narrated regarding istinja' with three stones, so perform istinja' with three stones and then do not mind what touches you from that water."