Your current circuit isn't even acceptable to use with a welder! That looks like 10AWG wiring, good for 30 amps(24 continuous like on an EV charging event), and you have a SIXTY amp breaker on it? Even the outlet itself isn't rated to handle 60 amps.
1) yes, you can use that wiring with a wall connector, dialed back to have a 30 amp breaker setting, and replacing the current 60 amp breaker with a 30 amp breaker. If you plan to use the UMC, you should replace the outlet with a 6-30 or 14-30, probably the latter.
2) its not allowed, but lots of people do lots of things that are not allowed.
3) you shouldn't be using the 6-50 right now. Its a hazard. It should be no more than a 14-30.