All signs point toward a hardware issue.
If your phone is under warranty, then take it back and have it repaired.
If its not, then I'd open it up and make sure all the connections are solid. if the phone was dropped its possible that a connector jarred loose. Also if the phone was exposed to any water or excessive humidity, make sure to clean your connectors with alcohol or something like that. Make sure to let everything dry before putting it back together and applying power.
If that doesn't work then you'd probably need a new LCD & Digitizer........



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