Does the obligation of the Udhiyah (sacrifice) fall away from the mother due to her illness? Is it permissible to donate its value on her behalf? And does she receive the reward for her brother's Udhiyah offered on behalf of their deceased father?

It is permissible for a person to slaughter a sacrificial animal (udhiyah) on behalf of the deceased, and the reward will reach them. If the one slaughtering intends the udhiyah for the household and the deceased, the reward will include everyone. As for the udhiyah from the money of a mother who has lost her memory, it is permissible to offer an udhiyah from her wealth, as udhiyah falls under expenses known to be good. It is also permissible for the children to give charity on her behalf from their own money. However, giving charity from the mother's own money is not permissible.