Is the game allowed to be ran on the hard disc drive, or does it have to be ran on disc when running on console?

Good question. No, Any form of USB or HDD loading is not allowed when running on console. Please refer to this thread for a general overlook of the rules before submitting a run: https://www.speedrun.com/series/spyro/forums/o05u8. Wish you the best if you do decide to run this game :)