Is it permissible to pay the salary of a teacher of the Noble Qur’an from money designated for a mosque, and is it permissible to grant incentive prizes to outstanding students from the same money endowed for the mosque?

It is obligatory to observe the specific purpose for which the donor intends their charity. If the owner of the money designated for the mosque, or prevailing custom, specifies a particular area (such as physical construction), it is not permissible to spend it outside of this area, like on Qur'an memorization circles or competition prizes. However, if its owner does not specify an area and there is no prevailing custom defining it, then there is no objection to spending it, or a portion of it, on what was mentioned in the question.