Should a metal box be grounded if there's no conduit connected to it?
You must never connect neutral to ground!!!

The situation of white and bare connected to the ground pin is a serious problem.  It means your electrical system now has two neutral-ground bonds, and that causes all manner of mischief.

I get it. If you've seen the inside of a main panel, you notice that neutral bar and ground bar are bonded to each other - or often, they are one and the same bar.  This can draw a novice to the conclusion that "neutral is ground".  Neutral is not ground.  That bond is a very particular thing, an equipotential bond, even if you see the lazy approach of using the same bar for both.  (I am no fan of this; I like it when I can clamp an ammeter around the neutral-ground equipotential bond. That is useful for detecting ground faults.) 

When an electrical system has two equipotential bonds, it creates a number of fairly serious "gremlins" in the system.  

The correct way is to cap off the neutral wire so it touches nothing -- and ideally, do this at both ends. 

Don't put 2 wires on a lug

...unless it says you can. I am assuming a NEMA 6-20 receptacle is not listed for 2 wires per lug, least of all a ground lug, since there is almost never a reason to need that.  (remember you are not allowed to use a device to daisy-chain a ground connection; doing so means if you remove the device, you sever ground for downline devices). 

Ground to the metal box first

The metal box should always be grounded.  If you need to ground 2 or more wires, then use a pigtail and wire nut.  

The receptacle may not need a ground wire

The receptacle has a metal frame or yoke that the screws go through.  If the yoke is clean bare metal, and it is attached to a box that is also clean bare metal, and the mounting screws are run down all the way (giving hard flush metal-metal contact from yoke to box).., with nothing in the way like a screw-capturing square of paper,... then you can ground the receptacle through the yoke.  There is no need for a ground wire.