American Dream and Promise Act of 2021 This bill provides certain aliens with a path to receive permanent resident status and contains other immigration-related provisions. The Department of Homeland Security (DHS) or the Department of Justice (DOJ) shall provide conditional permanent resident status for 10 years to a qualifying alien who entered the United States as a minor and (1) is deportable or inadmissible, (2) has deferred enforced departure (DED) status or temporary protected status (TPS), or (3) is the child of certain classes of nonimmigrants. The bill imposes various qualifying requirements, such as the alien being continuously physically present in the United States since January 1, 2021, passing a background check, and being enrolled in or having completed certain educational programs. DHS shall remove the conditions placed on permanent resident status granted under this bill if the alien applies and meets certain requirements, such as completing certain programs at an educational institution, serving in the military, or being employed. Furthermore, DHS and DOJ shall cancel the removal of certain aliens who had TPS, were eligible for TPS, or were eligible for DED status on certain dates. Such an alien shall receive permanent resident status upon meeting certain requirements and applying for such status within three years of this bill's enactment. For an alien seeking permanent resident status under this bill, DHS may (1) waive certain grounds of inadmissibility, or (2) provisionally deny the application if the alien poses a danger to public safety or has knowingly participated in certain offenses involving a criminal street gang. An alien shall have the right to administrative and judicial review of a denial or revocation of permanent resident status granted under this bill. An alien seeking judicial review of a provisional denial shall be appointed counsel upon request. An alien who appears to be prima facie eligible for relief under this bill shall receive a reasonable chance to apply for such relief and may not be removed until there is a final decision on that application for relief. DHS may not use information from applications filed under this bill or for Deferred Action for Childhood Arrivals status for immigration enforcement purposes. DHS shall establish a grant program for nonprofit organizations that assist individuals with certain immigration-related issues. This bill also repeals a restriction that bars a state from providing higher education benefits to undocumented aliens unless those benefits are available to all U.S. nationals without regard to residency in the state.