What is the ruling on directly teaching students outside the scope of the academy, benefiting from their money, and what are the implications of that?

The questioner is not at fault for what she did or what she intended to do. What is prohibited is working with students who are already enrolled in the academy or who have decided to enroll, due to the established prohibition of selling over one's brother's sale or bidding over his bid. This also includes leasing over his lease. If a lessor has agreed with a person, it is not permissible for another to offer more. As for someone who has not enrolled in the academy and has not decided to do so, there is no harm in working with them, based on the principle of permissibility.