When you lift the armature of the relay on the PCB ( without it's plastic case ), I would expect that the contactor will pull in ( clunk ), the welding torch tip & wire will go live ( i.e. you can get an arc ), and the machine would behave as if you had pressed the torch trigger. Is that the situation?
As for the fan, I don't know when to expect it to come on ( i.e. main power switch, or trigger press ), but anyway, are you sure the fan was working before the welder stopped working? Does the fan rotor spin freely if you try turning it by hand without power at all.... maybe it's just sticky.
No nothing like this happens