What is the ruling on intercourse after the cessation of menstruation for three months, and the discharge of a single brown spot once, followed by ritual bathing and intercourse four days after the brown spot? Is expiation required in this case?

There is no maximum limit for the period of purity according to scholarly consensus. Therefore, your wife must pray and fast, and you may have intercourse with her. The brown spot of blood is not considered menstruation, because the minimum amount for menstruation is a gush, and a spot or two spots are not counted as menstruation. She must make up for the prayers and fasts she missed.