Do brown discharges that sometimes appear at the time of menstruation after a hysterectomy, with the ovaries preserved, count as menses?

As long as your uterus has been removed, what you are seeing cannot be menses (hayd) or abnormal bleeding (istihadah). Rather, these brownish discharges do not necessitate ghusl (ritual bath). Instead, you should perform istinja’ (cleansing) from them and wudu’ (ablution) for prayer. If their discharge is continuous, you should perform wudu’ after the prayer time has begun and pray the obligatory prayer and as many supererogatory prayers as you wish with that wudu’. This is what Sheikh Ibn Uthaymeen, may Allah have mercy on him, mentioned. Therefore, a woman whose uterus has been removed does not have the rulings of a woman with istihadah apply to her. Rather, her ruling is like that of a woman who sees yellowish discharge, brownish discharge, or moisture after purity. She should not abandon prayer or fasting, and intercourse is not prohibited with her, and ghusl is not required from this blood. However, when she prays, she must wash the blood and perform wudu’ for prayer after its time has begun.