I am employed in the Institutional Equity office at a college, and believe I can help you with that one. The school (elementary, secondary, and post secondary) are required to provide an FM system for school and extra curricular activities. Not all FM systems work the same for all people, so they have probably ordered a model that they have used prior to see if it will be beneficial for you. You may require a different model depending upon the wiring in the building and your specific needs and level of hearing loss.
Yes, you can use the FM system without a HA, provided that you have earmolds. Or you can use headphones without earmolds. No, you are not responsible for paying for an FM system that is used educationally, but you will not be permitted to use it for anything other than activities related to school.
Hope this helps.