To stop the pop-up, Go to System Preferences: Security. Check the box next to "Require password to unlock each secure system preference." Then lock Security.
Safe Boot , (holding Shift key down at bootup), use Disk Utility from there to Repair Permissions.
Then move these files to the Desktop for now...
Reboot & test.
PS. Safe boot may stay on the gray radian for a long time, let it go, it's trying to repair the Hard Drive.
Claire, not sure if the Modem isn't connecting, or the Mac isn't connecting, but try it & report back.
Make a New Location, Using network locations in Mac OS X ...
Is that Interface dragged to the top of Network>Show:>Network Port Configurations and checked ON?