Can I deduct mandatory class fees under books, materials, and supplies? For example, to take a class I had to pay a set fee of $280, can I deduct this?

It is not deductible, but is a qualified expense for determining the Education Credits.

 In general, qualified tuition and related expenses for the education tax credits include tuition and required fees for the enrollment or attendance at eligible post-secondary educational institutions.

Yes, you can enter the fees that were not included in box 1 of the 1098-T, with books required to be purchased from the school. 

Alternatively, you can add then to box 1 of the 1098-T. Underneath box 1, In the 1098-T screen, click on the link "What if this is not what I paid the school" under box 1. You will then be able to enter your actual amounts.