What is the religious ruling on dealing with a father who commits adultery, drinks alcohol, openly neglects prayers, and rejects advice to return to God, knowing that his children are dutiful to him and hold a respectable position in society?

If the father commits great evil deeds and abandons prayer, then your duty is to be patient with him and supplicate for his well-being, and to continue to show him righteousness and kindness, and to advise him gently and kindly, striving for his guidance and repentance to Allah the Almighty, and seeking help from whoever you believe can influence him.