I completely agree with spitfire. The padding is probably the best idea. It not only keeps the proportions right, if done correctly it can illiminate the need for any strapping to hold it in place. A nice secure fit will allow you to simply slip on and go. Personally I've found cheap memory foam to work very well for this. And if you can find it try for the kind that transfers heat away from your body. That will help keep you cool.
Indeed it can! I personally like Egg Carton foam, even though it's warm, but that's just me. I plan on installing a battery operated fan inside my helmet at least to help keep airflow if I can.
Also, in reply to your signature. You should have had your students pep some pieces for you ^_^