As for keeping tank mates, given that this is a classroom and not a home, I would say don't do it. Normally I would say maybe depending on the fish but given the classroom and lack of experience (no offense,) I would strongly recommend keeping the betta alone..with maybe some cool snails
If you decide you want tank mates anyway, I'd stay away from any tetras and any schooling fish. Corys are usually okay to keep with bettas but you should have at least 6, and with the betta in a ten gallon, that's pushing it, and they really should have a sand bottom. A lot depends on your betta...bettas are such inquisitive creatures and their behavior and temperament widely vary. I am an advocate for saying that bettas can indeed have tank mates but under very specific conditions and limitations. I feel most hobbyists would actually recommend no tank mates what so ever, other than snails. A betta can get along great with other fish and snap at any point, even a year after no incidents have occurred. With your situation, the gravel tank, I assume it won't be "heavily planted,"classroom setting, inexperience, etc...don't risk it...my recommendation. If you had sand and a lot of plants, you could get the Pygmy corys.
You seem to be on the right track for cycling and what not. My opinion: you don't need to wait until the cycle is complete to add a betta in a ten gallon. You should indeed wait for a bit...add food like suggested several times every few days for a week or two, and then wait a week or two and add the fish. If adding tankmates, no matter the kind, wait until it's cycled. But then again tank mates should be added before a betta in my opinion to reduce the risk of betta becoming territorial right off the bat. Hope this helps. I'm no expert but I do have lots of experience in this matter and have done endless research myself. Many people will have different opinions based off of their specific experiences, and that's ok, many things in this hobby aren't carved in stone if you catch my drift. So many variables. Good luck and feel free to ask questions.