Is it permissible for a person who has repented from perversion, become diligent in prayer, and memorized some of the Qur’an, to lead people in prayer when the Imam is absent? And is it permissible for him to show his rectitude to people to affirm his repentance?

Congratulations on your sincere repentance which erases sins, along with the exhortation to remain steadfast upon it and to increase in good deeds, such as calling to Allah with sincere intention, and concealing past sins. There is no harm in leading people in prayer.