2. The competent department of driving school is the local transportation administration (committee), and they basically handle complaints. The police station may only deal with things like fraud.
3. If the competent department doesn't handle it, it's best to find media exposure.
4. In your case, it is estimated that it is difficult to defend rights without a training contract. You can complain to the Transportation Bureau and ask them to apply for a refund for you. Of course, there is a backlog of students driving in many parts of the country, and it is normal to take one subject every few months. If you want to get the certificate quickly, compromise and let the driving school arrange it as soon as possible.