Make the application for everyone after review?

The key the email is referring to is just your app's existing app key. While in development mode, only up 100 users can link your app to their accounts. Once in production mode, any number of user accounts can be linked to your app. The development/production switch doesn't affect how your app or the app authorization flow works.

It sounds like you just want your app to connect to your own account and only your own account. The API was designed with the intention that each user would link their own Dropbox account, in order to interact with their own files. However, it is technically possible to connect to just one account. The SDKs don't offer explicit support for it and we don't recommend doing so, for various technical and security reasons. (You also don't actually need to be production mode if you only want to use your own account.)

However if you did want to go this route, instead of kicking off the authorization flow, you would manually use an existing access token for your app, instead of kicking off the app authorization flow. (Just be careful not to revoke it, e.g. via https://www.dropbox.com/account/security .)