Answers

What file formats can I export?

You can export the data stored inside the feature layer in seven different formats: Shapefile CSV File KML Excel File Geodatabase (FGDB) GeoJSON Feature Collection

My organization would like to create our own dashboards.

Esri’s Operation Dashboards do not use templates per se. However, there are examples that can be found here. You will want to open one of these examples and Save As, effectively using the example as a base for your own dashboard.