When I perform an act of virtue, can I ask Allah s.w.t. to pass on (distribute) the reward to my living family (spouse, children, parents, brothers/sisters, etc.) as well?

Isaal-e-Sawaab (to perform a virtuous act and grant the reward to any person, alive or deceased) is permissible. On numerous occasions, Nabi (Sallallaahu Alayhi Wasallam) advised the Sahaaba (Radhiyallaahu Anhum) to give charity on behalf of the deceased.

One specific incident is of Hadhrat Abu Talha (Radhiallaahy Anhu) when the Aayat, 'You cannot attain virtue until your spend what is beloved to you.' (Nisaa 4). Abu Talha gave his most valuable well and garden, Bi'r Haa as charity on behalf of his deceased mother. In fact, Rasulullah (Sallallaahu Alayhi Wasallam) slaughtered 100 sheep during his farewell Hajj and made intention for himself and all those (alive, deceased and to come) who bring faith on his prophethood. (Nasbur Raaya, Allama Zailee). One may recite the Qur'aan, or any Dhikr or Du'aa and make intention of giving the reward to the deceased.

and Allah Ta'ala Knows Best

Mufti Ebrahim Desai