Is it permissible to register an apartment in the name of another person—a government employee—so that its buyer can pay its price in installments over 20 years, and are the monthly debt instruments registered in the employee's name considered a debt upon him during his life and after his death?

The aforementioned purchase and registration process is a fraud to unlawfully consume public funds, and it is not permissible to approve it, as it involves aiding in what is forbidden. You bear the responsibility for paying its price in this world and the Hereafter, as long as you agreed to register it in your name and caused the rightful owners to lose their right.