Fairness for High-Skilled Immigrants Act of 2020 This bill modifies requirements related to employment-based visas, changes annual limits on certain visas, and contains other related provisions. The bill increases the per-country cap on family-based immigrant visas from 7% of the total number of such visas available that year to 15%, eliminates the 7% cap for employment-based immigrant visas, and limits the percentage of employment-based visas which may go to an alien on an H-1B (specialty profession) or H-4 (family member of H-1B visa holder) nonimmigrant visa. The bill establishes various transition rules for employment-based visas such as (1) reserving a percentage of EB-2 (workers with advanced degrees or exceptional ability) and EB-3 (skilled and other workers) visas for individuals not from the two countries with the largest number of recipients of such visas, (2) reserving a percentage of visas for certain immigrants with approved backlogged petitions, and (3) allotting a number of visas for eligible professional nurses and physical therapists. The bill removes an offset that reduced the number of visas for individuals from China and also bars an alien affiliated with China's military forces from obtaining permanent legal resident status. The bill also imposes additional requirements on an employer seeking an H-1B visa, such as (1) requiring an employer to provide certain information in a new application for such a visa, such as the qualifications and application process for the position; (2) prohibiting an employer from advertising that the open position is only available to an H-1B applicant or that an H-1B applicant is preferred; and (3) prohibiting an employer of a certain size from having more than half of its employees as H-1B or other nonimmigrant visa workers. The Department of Labor shall create a publicly-available website where an employer seeking an H-1B visa shall post certain information about the open position to be filled by the H-1B worker. The bill also expands Labor's authority to review and investigate H-1B applications for fraud or misrepresentations of material fact, whereas currently Labor is generally only authorized by statute to review applications for completeness and obvious inaccuracies. The bill also increases civil monetary fines for certain visa-related violations. The bill also establishes a mechanism allowing an eligible alien (and the alien's dependents) to obtain authorization to work or travel outside of the United States, if the alien has an approved immigrant visa petition but is awaiting an immigrant visa to become available. Generally, this mechanism shall be available to an alien on an employment-based nonimmigrant visa who files an application with the Department of Homeland Security at least two years after the immigrant visa petition was approved.