He is not sure whether his father’s Hajj was valid; can he do Hajj on his behalf after his death?

Praise be to Allah.

Hajj is only required once in a lifetime, and the basic principle concerning its actions and rituals is that they are valid, so no second Hajj is required. But if you do Hajj on behalf of your father it will be a naafil deed, and there will be great reward in that for you and for him, if Allah accepts it from you. 

And Allah is the Source of strength. May Allah send blessings and peace upon our Prophet Muhammad and his family and companions. End quote. 

Standing Committee for Academic Research and Issuing Fatwas 

Shaykh ‘Abd al-‘Azeez ibn ‘Abd-Allah ibn Baz, Shaykh ‘Abd al-Razzaaq ‘Afeefi, Shaykh ‘Abd-Allah ibn Ghadyaan.