Yes, the ground needs to go to the metal box first. In most cases the socket will pick up ground off the metal box and no ground wire is needed. The conditions for that involve a receptacle marked "Self-Grounding", or hard flush metal-on-metal contact between receptacle yoke (metal frame) and receptacle box.
The metal box will either have a ground screw, or a site designed to accept a machine screw (bolt) threaded #10-32.
For a 50A circuit using THHN wire in conduit, and with 75C terminals at the outlet (most are)... you need #8 copper conductors.
For a 50A circuit, you need #10 ground with your #8 conductors.
You need a #8 ground, actually.
Whenever you use larger wire than is required for the conductors, you need to enlarge the ground in proportion.
#8 would suffice for your conductors and you chose #6 instead. Since you bumped the conductors by 2 wire sizes, you must bump the ground by 2 wire sizes as well. So #8 ground with your #6 conductors.