when a parent outwardly professes Islam but his son has unresolved doubts about possible nullifiers, may the son continue to apply the outward ruling of Islam for inheritance unless apostasy is actually established?

Yes, Islam established with certainty is not removed by doubt, so one who outwardly professes Islam keeps the ruling of Islam, inheritance included, unless apostasy is actually established. Doubt about a nullifier does not establish it, and no particular man is ruled upon until the conditions of takfir are met and its impediments are absent which is the duty of a Muslim Judge.