Is it permissible to join the Faculty of Medicine through favoritism (wāsiṭah), and what is the ruling on the money earned from working after graduation using that favoritism?

If the available places in the university are limited, and your admission through "wasta" (connections/nepotism) would deprive a deserving student, then your enrollment is impermissible; because it is an injustice and an infringement upon the rights of others. However, if your admission will not lead to depriving deserving students, and "wasta" is prevalent in society, then there is no harm in it.