Is it considered a forgery to obtain a certificate of residency through unofficial means and without a lease agreement, and is it permissible to use it to obtain employment?

If you are already a resident of the state, there is no harm in obtaining a certificate that proves this reality, even if it is through influence or mediation; because it falls under the category of good intercession by which people's needs are fulfilled, provided that it is limited to reaching your right without encroaching upon anyone else's right. This type of intercession is commendable in Islamic law.