When applying for a patent, the inventors are me and my mentor, and the patentee is our school. Who does the patent belong to and who has the final transfer right?

You and your tutor are inventors, and it should be a service invention. Therefore, the patent right belongs to the school. However, if the results of both of you are not service inventions, you and your mentor can become patentees. But since you have said that the patentee is the school, then of course the transfer right also belongs to the school. Only the patentee has the right to transfer the patent, not the inventor. The inventor only has the right to put his or her name on the patented product.