Is it possible to buy a licence with my personal email then use the gradebook with my school email?
Yes if you change schools you can change your license email as well to the new school.
Excellent. To follow up, if I changed my licence to my school email, would I eventually have to change it back to my personal email in the case that I change schools?
You can buy the license using your email but activate the Gradebook for the school email using this form https://docs.google.com/forms/d/1cERJbWF1hv7PkiUVUqXUXzPR6GsX_JmybmoZr9iIKTU/viewform?edit_requested=true