District of Columbia Courts Improvement Act of 2022 This bill amends certain laws relating to the District of Columbia (DC) courts. The bill revises and modifies the rules and procedures for the service of jurors in the DC courts, including to allow the exclusion from jury service of an individual who is 70 years of age or older upon the person's request. Grand and petit jurors serving in the Superior Court shall receive fees and allowances at the same rates provided for fees and allowances paid to grand and petit jurors appearing in the district courts of the United States. The bill adjusts the amount of compensation paid to attorneys representing indigent defendants in criminal cases and criminal investigators in DC courts to equal the amount that is paid to such persons with regard to a U.S. district court. The bill also modifies the process for appointment of the Register of Wills; authorizes retroactive pay adjustments for employees of the DC courts; revises certain administrative functions, including those concerning use of fees, acceptance of gifts, procurement of motor vehicles, and use of space in courts buildings; and revises the administration of the Small Claims and Conciliation Branch, including regarding the use of arbitration, mediation, and conciliation to settle cases.