You could probably find charts online that have the diameter for each AWG size, and resistance for a given length... I got it outta one of my older EE books, but I figure you could find it all online just as easily. Its also possible to figure out the resistance if you know the resistivity and the length, but it's just so much easier to look at a chart, LOL.
You should be able to get the maximum current draw from Lincoln... If you know that, you can calculate the transformation ratio, which you can then use to figure up the current the welder would need to give a specific output, or figure what the maximum output is for a given max current input.
What D. Hearne is talking about with current traveling on the outside layer of a wire is called the skin effect. This is true, but it is dependent on the frequency, and at 60 Hz it shouldn't be a real issue (although I hate solid wire, not only b/c of skin effect but its just a general PITA to work with when you start getting up in gauge). It's caused by the wire's self-inductance, when the changing current of AC flows through the wire, it creates a magnetic field that induces a voltage that opposes the flow of current.
Transformers also use inductance caused by the changing current, but they use an actual magnet with wire wound around it to induce a voltage in another wire thats wound around the magnet, where the number of turns is directly related to the output voltage. Thats how the voltage selector on your welder actually works, it only uses so many of the available windings to induce a lower voltage than if it used all the available windings.
AC is really neat in how easily you can manipulate it, it would suck if back in the olden days they woulda decided to use DC instead of AC in our houses. I could go on for hours, AC is just a really great thing.
With the 30A breaker, you are just limiting your input current which limits the output current that the transformer can induce on the output side... This won't hurt the transformer, just limit your max output. Like I said, if you get the max input current the welder needs to make its max output current, the transformation ratio can be figured out and then you can figure how much input you need for a specific output current and so on. Transformers are really amazing, considering how useful something so simple really is.
I like your idea of 3 ground wires and 2 hot/common wires, you can't overground. Safest way of doing that.
My buddy has the same welder and he got some 7 strand trailer wire (.60 cents a foot) and trippled up 12 gauge wire and the remaining 14 gauge was used for the ground fault. 
In my shop I have a 220V receptacle right next to my breaker panel, which was an easy install.
