No, it's a single center tapped 220V supply.
Yes and no. You need two wires to get your 220V, you need a third wire - safety ground as a code requirement. Neutral can be provided (it's the center tap), but it's not necessary unless you also need access to 110V which a welder does not need.
Two ends of the center tapped 220V utility transformer secondary.
Absolutely there is, it is the center tap of the 220V transformer secondary winding. The neutral is your reference when you need 110V.
Well, close to it at any rate depending on the quality of your grounding rod(s), soil conductivity, fault currents, ground loops, etc.
Er, not really. You are utilizing the 220V secondary of the distribution transformer. When you are using only 220V the existence of the center tap (neutral) of that transformer secondary is irrelevant.
Again, there is not a 110V line in reality, there are two legs of a 220V transformer secondary and a center tap that is referred to as the neutral. The neutral (center tap) is bonded to the ground at the service entrance and nowhere else. The neutral normally runs at pretty close to ground potential.
The old (and grandfathered) dryer outlets were two hots and a ground (not neutral). The old code allowed a small 110V load to be placed on the ground conductor (the dryers timer and sometimes motor).
Not quite following you there. For a 220V welder, all current is on the two hot leads, there is no neutral connection at all and the ground is purely a safety ground and carries no current unless there is a fault.
Not really.
Pete C.