On the dashboard, you can add an Alert Reports widget that has a few options from the dropdown.
Is it possible to have this same kind of widget but I can add my own reports to it, and specify the category of each report (high priority, important, informational)?
Yes you can. When the dash board opens, click config in the bottom right of the widget. In the next screen you will see all the dashboards. In the dashboard that you want to add the report to, choose add report in the High priority, important, or informational section. You will be shown a list of all reports. Be carful of what reports you add. The more reports that are added the longer the dash board will take to load.