Okay, the first thing to do is check out the welder and see if it is a 3 wire or 4 wire equipment. If the cord does not have the required number of conductors, replace it with a properly sized cord with the correct plug on it. Replace the receptacle with a receptacle that matches the plug. Check breaker sizes and ratings of the cord, circuit conductors and plug receptacles to ensure they are properly sized and protected.
Now, if it is a 3 wire, and third wire is bonded to the metal case of the welder, this should be a ground, newer welders should not use this as a neutral. Neutral and ground are not the same. Older welders may do this, but it is not an ideal situation.
Answer this for me, if you are using the ground, that is bonded to the metal case for a neutral for a 120V circuit, there will be some current on that ground/neutral, including the case. Wouldn't this cause the case to be energized, thereby shocking someone who touches it?
yes, this can happen, I've seen instances where a neutral was incorrectly bonded to ground out in a circuit, this backfed current through the ground system, and the person's kitchen sink was live, they got shocked everytime they touched it.
I believe the idea on dryers using 120 to run the timer is false, If I remember correctly, there is a stepdown transformer, 240VAC to 12 or 24VDC. I will have to look at my dryer a little closer to be sure.
When I am preparing plans for projects, if I am feeding a 240V load, it gets 2 hots and a ground, not a neutral. If it is 240V, with 120V control, it either gets 2 hots, a neutral and ground, or it gets 2 hots and a ground, and the 120V control is fed by a separate circuit with 1 hot, neutral, and a ground (the ground may be shared between the 240V circuit and 120V circuit only if the circuit conductors are ran in the same conduit or multiconductor cable.
Any questions, please ask away. If you need help sizing conductors, breakers, etc. let me know, I'll give you a hand.