Is it permissible for a pilgrim residing in Jeddah, coming from Makkah Al-Mukarramah with the intention of performing Hajj, to enter into a state of Ihram from his family's home in Makkah, and does he then have to perform the Farewell Tawaf?

Since you reside in Jeddah and your visit to Makkah is not considered a residency, you were obligated to enter into ihram (definition) from Jeddah. The Farewell Tawaf (definition) is one of the obligatory rites of Hajj. By omitting these two obligations, two 'dams' (sacrifices) became incumbent upon you. A 'dam' is a sheep to be slaughtered in the Haram (definition) and distributed among its poor. If you are unable to do so, then you must fast three days during Hajj and seven days when you return to your family for each of these two omissions. If you had performed the Farewell Tawaf, then only one 'dam' is incumbent upon you.