Could an integration live on the dashboard upon a user downloading the integration? Is there a standard to how an integration works/lives on the dashboard? Is there an example we could see? Currently, the dashboard just sits on the loading screen in the developer account, is that because we do not have a site hooked up? I can still get a screenshot of a dashboard though, I just would like to see a live example at some point too.
Are you asking if an integration could appear on a HubSpot in-app dashboard? This isn't currently possible; the only way for an integration to appear in-app is using the CRM extensions and/or Timeline APIs. I can totally see the use case for that, though; If you have the inclination, I would encourage you to check out the Ideas Forum on the HubSpot Community. There, you can create a post including your use case that product and other customers can see.
As far as Timeline APIs go, if we were to integrate with them where would the integration appear in-app? Productivity?
So if you create an app that uses the Timeline API, the events you create would appear on the timelines of CRM objects (contacts, companies, deals, etc.). This API allows you to create custom events that add your apps info to existing records, but it doesn't create any dashboard for your app within HubSpot. The API closest to providing that functionality is the CRM Extensions API: