I didn't know that being a cheer coach would be considered self-employed so some of these questions I'm not sure how to answer

I coach at the school I teach at; so do I use the school's address?

You can use the school's address or your address.

As far as a name do I use the organization name?

For name, you use your name.  You are the self-employed individual.

Can I claim the ribbon I purchased to make the girls bows and the snack and treats I also purchased?

Yes, you can.