As you explicitly pinged me for an answer:
AFAIK, apps can only use permissions declared in their Manifest – so there should be no way an app could add itself permissions it never requested there.
DeviceAdmin gives high extra powers (apps are e.g. protected against uninstall, and can access some extra stuff they couldn't do without). What exactly this includes, I'm not sure. I only can refer to a specific part of the page you've linked, where the powers are explained. I cannot vouch for the pages completeness (or correctness at that: wouldn't be the first time some error made it in there).
Enabling Device Administrator (source: Android Developers; click image to enlarge)
You might remember there's something else carrying a similar name: ADM, the Android Device Manager. Checking above screenshot with that in mind, gives you an idea on the basics: ADM allows you to permit those functions (though AFAIK it doesn't feature "Monitor screen-unlock attempts" and "Disable cameras"). But this is a typical example of using most of those features.
TL;DR: AFAIK DeviceAdmin cannot grant permissions to any app (including the DA-app itself), especially not if the app-in-question did not declare them in its
Manifest. It just gives the app carrying this permission access to specific system features not available otherwise.