IPhone – Password lock for shut down screen

This indicates a hardware problem with your phone. Contact AppleCare, or make a Genius Bar reservation at your local Apple Store

Lost mode combined with iOS 7 is extremely useful as long as th phone has cellular service or is connected to WiFi networks that it trusts. The device needs to connect to the Internet to be re activated after wiped, at which point it will enter lost mode and refuse to work again without your password for Apple ID.

Be sure to not remove that device from your account since that makes the device far more risky and expensive to sell, pawn or try to use by the thief or even someone that ends up with the device. Someone could desolder the CPU and try to reserialize or cannibalize the parts, but the quick hit of cash for a stolen device is defeated by Activation Lock and Lost Mode features.

The device shows as offline when it hasn't checked in recently to the Apple servers despite an attempt by the Apple Push Notification service attempting to send a check in request to the last active address of the device on the Internet.

If the device ever gets connected to Wifi or cellular data reconnects it, your Lock request will get sent, acknowledged and you'll get an email. If you are quick, you can log in to find my friends and see if the location data updates as well at that point.