Are the quiet musical backgrounds in purposeful educational videos considered among the forbidden musical instruments? And if so, how can one dispose of the money earned from these works?

Music is among the forbidden musical instruments (ma'azif) according to the majority of scholars. Therefore, it is not permissible to include it in videos. One can use sound effects and permissible rhythms instead. Money earned from designing something permissible is considered permissible, even if it occasionally included music due to not knowing its prohibition, and because that was not the basis of the contract. What is tolerated in a subordinate matter is not tolerated in a primary matter. Nevertheless, it is obligatory to refrain from including it in the future.