Context
We are a company that builds a live chat solution. Recently, we decided to save the conversations into our customer's Salesforce accounts. We did this by implementing a connected app to which our users would login with their Salesforce via oAuth. After we retrieve their access token we start creating in his name an activity for each call.

Questions
Is it ok to have this app on a developer edition account?
Should it be published somehow, in the AppExchange or somewhere else?
Will other accounts, rather than development edition ones, be able to log in into it?