American Housing and Economic Mobility Act of 2021 This bill addresses housing affordability and availability, makes certain changes to the estate and gift taxes, and contains other related provisions. The Department of Housing and Urban Development shall provide grants to (1) state and local governments that remove unnecessary barriers to building affordable units, (2) states to assist borrowers who have negative equity in their homes, (3) state housing finance agencies to construct or acquire affordable rental housing and prevent tenant displacement and harassment, and (4) eligible individuals (generally, lower income individuals who resided in low-income or historically racially segregated areas) to help them purchase their first homes. The bill provides funding through FY2031 for the first-time homeowner grant program established by the bill, through FY2022 for the affordable rental housing construction program established by the bill, and through FY2022 for specified existing rural housing programs. The bill also expands fair housing protections to prohibit discrimination based on sexual orientation, gender identity, marital status, source of income, veteran status, or an individual's perceived membership in a protected class (e.g., an individual's perceived race or sex). Additionally, the bill modifies the estate tax in various ways, including by reducing the number of brackets to three, increasing the tax rates, and reducing the basic exemption amount. The bill also modifies provisions related to gift taxes and transfer taxes.