Is the aforementioned work permissible in teaching students, and receiving money in exchange for it?

As for the money, there is no harm in it, because it is a fee for permissible teaching. As for utilizing other websites that provide exams not available on the Ministry's website, it is not permissible to attribute all of them to the Ministry; rather, it is necessary to indicate the multiplicity of sources. If these websites are public and freely available to everyone, then there is no problem. However, if they are private and reserve copyright and sell exams, then it is not permissible to take them without their permission.