Yes, if you implement that. geojson.io currently works with GitHub - you would need to implement a new source. This is an exercise for the reader - you'll need to figure out how to do this on your own.
Yes, if you set up your own gatekeeper instance and configure your instance with your own api keys. Like the previous question, you will need to figure this out.
There is none - geojson.io is a static application that's hosted on GitHub Pages, and could be hosted anywhere else, as static files.
There are none - geojson.io bootstraps its entire user interface with JavaScript and d3. It does not use templates.