Funding Attorneys for Indigent Removal (FAIR) Proceedings Act This bill expands access to counsel for non-U.S. nationals ( aliens under federal law) in immigration proceedings and provides protections related to such proceedings. Such an individual 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 individuals in any immigration proceeding, whereas currently DOJ is statutorily authorized to provide counsel only for 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 individual's removal shall be stayed upon the motion's filing. The bill establishes the Immigration Counsel Account to be used to provide counsel under this bill. A portion of certain immigration fees shall be deposited into the account. The Department of Homeland Security (DHS) shall provide each individual, generally at the beginning of proceedings, a complete copy of the individual's case file in DHS possession, unless the individual waives this right in writing. Failure to provide such documents shall delay the individual's removal proceeding. DHS shall facilitate for all detained individuals access to counsel and programs that inform such individuals of their immigration-related rights and obligations.