outlet not working but grounding screw is hot
Rewrite

It's not metal conduit, those are Romex clamps.  

The first problem is that you have power on the receptacle somewhere when you expect not to.  So keep using that voltage tester.  There are lots of ways to use a voltage tester wrong, but this for sure: it should never light up on a dead circuit. if you have turned the circuit off, and it is not dead, it is now very important to find out which other breaker you must turn off to make it actually dead.  It's likely these two breakers together form a multi-wire branch circuit, which share a neutral in an engineered way which works. However there are certain rules MWBCs must follow, and if they do not, they become nightmares, with symptoms like these.   We need to know if 2 breakers feed that circuit so we can work that issue... if it's an issue.  

Looking at that melted neutral, it would also matter to know exactly where in the service panel those two breakers are.   That super matters.  

It would also be good to know if this installer was habitually using white for hot and black for neutral. (Maybe he was British?) 

It would also be good to get a length of wire so you can get a reliable ground from some place like a sink, spigot, water pipe etc. 

Aside from that,  screws shorting against the metal box could contribute to the problem.  Wrap the receptacle with tape edgewise to stop that from happening.