Yes, connecting the welder's ground to the "neutral" wire is correct. The dryer plug is probably a 3 prong model with a "neutral", but no "ground". In this situation, the difference between ground and neutral is inconsequential. I think you have it right. The green goes to the center terminal, the white and black go to the outside terminals, and the red is just taped off and put to the side.
You can use http://powerstream.com/Wire_Size.htm to compute the voltage drop at a length of run of a certain gage of wire. It has a nice JavaScript calculator at the bottom (that I used for the info I posted above).