What is the ruling on creating a religious group on Facebook as an ongoing charity (Sadaqa Jariyah) for the soul of a deceased parent, and what is the optimal way to deal with the dissemination of weak or fabricated hadiths by some members?

There is no harm in keeping the account active and monitoring it to spread goodness, authentic hadiths, and Islamic rulings, with the caveat not to publish incorrect information. You are not sinful for what is published without your knowledge, for Allah, the Almighty, says: "And every soul earns not [consequences] except upon itself, and no bearer of burdens will bear the burden of another." If you become aware of incorrect information, it is obligatory to delete it or to issue a warning about it. As for your father, he is not sinful because his responsibility has ceased, and he will receive reward for the good that was published on the site, for the Prophet, peace and blessings be upon him, said: "When a person dies, his deeds come to an end except for three: ongoing charity, knowledge from which benefit is derived, or a righteous child who prays for him."