Is it permissible to inform others about sponsoring an orphan and how life changed for the better after the sponsorship, as a way of encouraging them to do good deeds? And if it is not permissible, what should one do?

There is no harm in disclosing the sponsorship of an orphan if the intention is to encourage others to do good or to speak of Allah's blessings. A sincere deed is not harmed by disclosing it for good purposes. However, it is better to conceal voluntary good deeds to preserve sincerity and avoid ostentation (riya'), as Allah Almighty says: "And if you conceal them [i.e., your charitable donations] and give them to the poor, it is better for you." If the disclosure is out of sincerity, its doer is rewarded. But if it is for ostentation, it is forbidden and nullifies the deed, and repentance is required for it.