Fair Day in Court for Kids Act of 2021 This bill provides legal protections for a noncitizen (i.e., an individual referred to as an alien in federal law) who is detained or subject to immigration-related proceedings. The Department of Justice (DOJ), or the Department of Health and Human Services (HHS) in a case involving an unaccompanied child, may appoint or provide counsel at the government's expense to noncitizens in removal proceedings and related appeals. An unaccompanied child must be represented by counsel paid for and appointed by the government at every stage of such proceedings unless the child has obtained counsel at their own expense. If HHS fails to provide counsel to an unaccompanied child, the deadline for filing a motion to reopen a proceeding shall not apply, and the filing of such a motion shall stay the child's removal from the United States. The Department of Homeland Security (DHS) must provide a complete copy of a noncitizen's immigration file to the noncitizen (or the noncitizen's counsel) within seven days of a notice to appear for an immigration proceeding, and failure to provide the file shall result in a delay in the proceeding. DHS must provide access to counsel for all detained noncitizens. The bill authorizes HHS to enter into contacts or award grants for providing immigration-related legal services to children. The Office of Refugee Resettlement must develop model guidelines for representing noncitizen children in immigration proceedings.