Well for starters, turn the phone off, then pull the battery out for 30 seconds. While you have the battery out pull the sim card out then reinstall it also. Turn the phone back on, that might clear things up.
I've tried that a few times but to no luck.
You can try to force the phone into a factory reset turn phone off, (not always required) Pull Batter, Put Batter back in, Hold VOL Up and PRESS and hold power till the Boot menu comes up select Factory reset (Vol up and Down to select) and press the center home button (Or power), Confirm and watch and see....
If there is a option to Clear Cache you can try that before the Factory Reset.....I can't remember if that option is on that screen...
Here is the proper Procedure....
Turn off the device
PRESS and HOLD: HOME KEY, VOL-UP and POWER Key
When the Phone Vibrates, let go of the power key but continue to hold Vol-UP and HOME Keys
When the Android System Recovery Screen appears, Release the Vol-Up and Home Keys.
Press the Volume down Key to highlight 'wipe cache partition'
Press the Power Key to Select, The hone should automatically restart of the cache is wiped.
done all of this still stuck !
Sounds to me like the OS is corrupted - I would do like BakerGeckos has said and just do an FDR
With what you have said and the problems you are having I would just dump and start fresh - Sorry
When you FDR it and do the system updates do a backup with Samsung Kies 3
Again I'm sorry I have to say do an FDR, but without me holding it and all I see you said and tried this is all I can safely tell you without turning it into a brick.
I've been in IT for over 25 years and been on Android since Beta days
Yeah Sounds like a Factory reset which is in that same menu......
I had the same issue this morning. Factory reset did not work. I'm getting a new phone shipped to me tomorrow. It's frustrating because they have S4's in the store but because this is under warranty they have to go through the warranty dept. Nice service Verizon.