Job "hired me" but no response... what do I do?

I think you did the right thing by emailing the company back. If I were you, I would look into the company and its background. Something seems a little shady, however it could just be a lapse in communication. If you do not here anything by next Tuesday, I would email again and explain that you are confused about the process, and that you are still interested in the position, but are anxious to get started.