OK, that's excellent.
The next step is to turn off the welder, remove the brown and brown/orange wire (6 and 7) and use a multimeter to measure the resistance across these spade connectors.
It should be a complete open circuit, but I'm guessing it won't be.
Now I'll show my lack of understanding. When you trigger the MIG gun, I presume it is supposed to do more than just turn on the wire feed motor. Does this provide power to whatever provides the arc as well? It certainly sounds like that from your description of what is connected to the brown/orange wire.
My guess is that there *could* be some metal swarf around the base of the relay, but you indicate that there is also sparking within the relay, so that doesn't fit. For the relay to be sparking inside, the voltage must be way higher than the rated voltage for the relay. I'm wondering if there's not some other fault.
If you were brave, you could try triggering the welder while the relay was sparking (preferably with all the covers on in case something explodes) and see if the welder does what it's supposed to. In theory, the relay closing should eliminate the sparking, but I'm not sure if it will ever open again -- have someone manning the power switch. <-- this will tell you a lot, but I'm not sure it's the best idea in the world.