How can I make a user-friendly account approval for admins?
There are two modules that you can take a look into 

Admin Users


  This module helps to administrate all registered users. You can sort
  out users according to their status (active or blocked), the number of
  published nodes, the number of comments, the last visit or the period
  of registration. You can also change the role of any user, block or
  delete many users with one click just ticking them in the table.


Multi-admin pending approval notifications (MAPAN)


  Problem
  
  The current behavior in Drupal core when a new user registers
  requesting approval sends mail only to the site's contact address.
  Thus only one person is notified even though there may be several
  administrators/moderators.
  
  Solution
  
  This module improves this basic behavior by notifying all users that
  have the 'administer users' permission by default.
  Administrators/moderators get a setting to disable these notifications
  in their user profile form. A link is also appended to each of these
  notifications.