Is it permissible to give charity on behalf of a father who does not fast and smokes heavily? And is one who abandons prayer considered a disbeliever, especially since he suffers from a psychological condition and refuses to perform acts of worship?

Your father needs to see a psychiatrist, a trustworthy person who treats with the Quran and Sunnah, and a righteous person to advise him, along with increasing supplication for his healing. He should not be excused from prayer or fasting as long as he is of sound mind and capable of performing them; feeding the poor is not a substitute for him except in cases of continuous inability and with his permission. Abandoning prayer is a grave danger and a great sin, but he still remains within the fold of Islam. His guardianship over his daughters is not valid, and the matter should be referred to a judge to rule on it and appoint someone suitable for guardianship.