Replacing a wired remote (potentiometer) with wireless remote. Complete Newbie

I became interested in arduino a couple of weeks ago. Prior to that my electrical knowledge consisted of knowing the pointy end of the battery goes to the front of the flashlight and associated experiences getting shocked while welding.

I am interested in building a wireless remote control for my welding machine. I have made it through 5 or 6 of the 1st sketches.

What I am curious about is this

1) What wireless communication distances can be achieved with arduino and associated shields or other hardware. 2) Is there a likelihood of interference with the close proximity of a 480v powered welding machine that is an inverter type power source for welding. 3) What is the replacement for a potentiometer that can controlled digitally.

4) The input on the potentiometer is 15 volt. Is that input voltage controllable and what do I need to study up on for controlling items powered from other sources.

Thanks in advance and forgive any poorly phrased questions.

Gerald Austin