You certainly have done great troubleshooting so far.
It is always possible that it could be hardware and assuming you have no water or accidental damage your limited warranty should still cover you and you could get it repaired at an apple store, apple authorized service provider, or call apple and get a mail in repair. THOUGH before that try these steps.
1)Try going to Settings/General/Reset/Reset All Settings.
Reset all settings should help a lot with this type of issue.
2)Next you will want to plug your device into a computer and back it up, Then from itunes tap "Restore Iphone", once the device is running again try setting it up as new and confirm if the fingerprint scanner is having an issue. If it is having an issue still it is 100% hardware and your information is backed up and wiped from the device which the store will want. If it works then restore your backup to the device.