What permissions are granted to an app with device administration privileges? Does a Device Administrator application have any/all permissions such as contacts, storage, etc.. ?

It's not clear from developer.android.com device-admin whether or not these admin rights also imply application permissions.

Or does device administrator simply mean that it can only do the things in the policy it's allowed to such as lock the screen, wipe the phone, prevent (easily) uninstalling the app, etc.. and that permissions to access private information is managed via the permissions settings only ?
Yes, a device administrator only has permissions to do what it says.

If you change the permissions of the app that the administrator is associated with, the administrators permissions will not change

(Answer made from the following sources):