Funding Attorneys for Indigent Removal (FAIR) Proceedings Act This bill amends the Immigration and Nationality Act to authorize the appointment of counsel, at government expense, for aliens in immigration proceedings and to automatically provide to such aliens a complete copy of the their immigration file in the possession of the Department of Homeland Security (DHS). DHS must also facilitate access to counsel and legal orientation programs for all aliens in DHS detention facilities. The bill also establishes the Immigration Counsel Account to collect fees for providing access to counsel when required or authorized by this bill.