Hello,
Integral is using FontAwesome version 4.7.0 and we will be upgrading to version 5.0 very soon.
Here is the cheatsheet for available icons for version 4.7.0
https://fontawesome.com/v4.7.0/icons/
Regarding the inability to upload your own icon, you can try navigating to another page with the Admin Dashboard and then going back to the Widgets page, then try and click the upload button.
Let me know if that doesn’t work.