Is a Farewell Tawaf (circumambulation) obligatory for the wife and children, given that they reside in Mecca and will travel ten days after performing Hajj, and is a sacrifice (hady) due from them if they perform Umrah in Dhul-Qa'dah?

Your wife must perform the Farewell circumambulation (Tawaf al-Wada') if she intends to travel after Hajj, unless she is menstruating. If you are residing in Mecca for work, then you, your wife, and your children are not required to offer a Hajj Tamattu' sacrifice (hady) if you performed Umrah during the months of Hajj and then performed Hajj in the same year.