Funding Attorneys for Indigent Removal (FAIR) Proceedings Act This bill expands access to counsel for aliens in immigration proceedings and provides for protections related to such proceedings. An alien shall have the privilege of being represented by counsel in any immigration proceeding, whereas currently this privilege is statutorily provided for only in removal proceedings. Similarly, the Department of Justice (DOJ) may provide counsel at government expense to aliens in any immigration proceeding, whereas currently DOJ is statutorily authorized to provide counsel to aliens only in removal proceedings and at no expense to the government. Under this bill, DOJ shall provide counsel to children and vulnerable individuals in such proceedings, at government expense if necessary. If DOJ fails to do so, certain limits on filing a motion to reopen removal proceedings (such as the time limit for filing) shall not apply, and the alien's removal shall be stayed upon the motion's filing. The bill establishes the Immigration Counsel Account and a surcharge on various immigration fees. The surcharge shall be deposited into the account to be used to provide counsel under this bill. The Department of Homeland Security (DHS) shall provide each alien, generally at the beginning of proceedings, a complete copy of the alien's case file in DHS possession, unless the alien waives this right in writing. Failure to provide such documents shall delay the alien's removal proceeding. DHS shall facilitate for all detained aliens access to counsel and programs that inform such aliens of their immigration-related rights and obligations.