Is it unlawful for me to keep a copy of the Qur’an that was gifted to me by someone who believed I helped him return to service, while in fact I did not, and is it permissible to donate it to a mosque?

It is not permissible to take a fee for intercession, for it is the zakat (definition) of one's prestige. The Prophet, peace and blessings be upon him, encouraged intercession without compensation. Therefore, you must return the gift to its owner. It is not permissible to place it in the mosque except with his consent, after informing him of the Shar'i reason for not accepting it.