Should the congregation expose the muezzin who falsified a certificate of memorizing half of the Quran to obtain his position, and who leads them in prayer despite the presence of others more حافظ (knowledgeable of the Quran by heart) and more learned than him?

Falsifying a Quran memorization certificate and using it to gain employment is forbidden. Officials must appoint those qualified for the position, for appointing the unqualified is a betrayal of trust. The congregants of a mosque are permitted to demand the appointment of an Imam and a Muezzin who meet the Shariah requirements.