iPhone 5c Digitizer or logic board damage
If your phone rings when called, that tells you a couple things that the phone is able to do:


The phone is able to turn on.
The phone is able to process the received call (and I'm guessing all of iOS's pretty effects with loading it as well).
The phone is able to process events from all of the phone's physical buttons (the Lock button, Volume Up/Down, and the Home button).


However, as it seems, the phone isn't able to send touch events to the UI, meaning the digitizer is broken.

Screw in the loose screw, replace the screen/digitizer unit, and reboot. All will be well. :)