The calculator you referenced is only appropriate for voltage drop and makes the disclaimer that the results of the calculation do not account for current specs or NEC. Any circuit fused for 30 amps must use a minimum of 10 ga copper or 8 ga alu. Longer runs may require an upgrade of wire size. In your case, use at least 10 copper for your welder regardless how far it is from the breaker panel.
I'm sure one of my buddies here will have a NEC ref for ya.
There's a decent discussion about making aluminum wire connections over here: What's the best way of replacing a plug or switch in a house with aluminum wiring?
If there is #12 Al wire on the circuit, the breaker should be 15A. #10 aluminum wire can have a 20A breaker, and you shouldn't normally see anything like #14 aluminum. Or you could just pull new copper wire and permanently disconnect the aluminum. Your home insurance company might not even cover you if they discover that you have aluminum wiring on your branch circuits. Aluminum is still common for the big feeders into your main panel, though, and that's fine because it's terminated appropriately by the electrician.
Any device that aluminum wire is connected to should be approved for connecting to aluminum wire, and any connections between aluminum and copper wire have to be made with special connectors. There are different types. Some are crimps that compress the wires very tightly to "cold weld" them together, some are special wire nuts, and you must use aluminum oxide inhibitor. The oxide inhibitor is included in the special purple wire nuts that are rated to connect aluminum and copper.
One of the issues is that aluminum and copper have different rates of thermal expansion and contraction. Combine that with the tendency for corrosion when aluminum and copper are in constant contact, and the fact that aluminum oxidizes (rusts) very rapidly if oxide inhibitor isn't used. All the inconsistent expansion and contraction and corrosion helps to loosen connections. Add the fact that aluminum oxide is an effective electrical insulator and you have a lot of potential for connections to go bad over time, leading to overheating and fires. Loose electrical connections get hot (and might also produce sparks), and heat starts fires (as do sparks, since they're hot), etc.
If you're going to leave the aluminum wire in place, though, maybe you want to avoid the twist-on connectors (wire nuts) and use the cold-weld crimp-on connectors, then seal the connections up really well with something like lineman's rubber splicing tape.
You don't want this in your walls:
Best Answer
Putting this trick first but it is my last resort.
To find wiring when I don't have my scanner, a simple stud finder with a voltage alert. Turn all the other circuits off then using the stud finder watch for the voltage alert to light up. You can get within a couple of inches of the wire path through the walls and figure out where it goes. I have used this method several times where I was not planning on doing any electrical work. It is cheap and works well but you do need to turn the other breakers off for this method to work well.
To answer your question, here are my normal methods to trace wire and find circuit breakers.
I use a Greenlee CS-8000 it injects a signal on the wire and can trace wires through walls and even conduit. This tool is expensive ($995) and not worth it for homeowners.
When I am doing telecom I use a less expensive toner/tracer again made by Greenlee but there are lots of low voltage toner/tracers both of the above can be used with no power on the circuit or live. (The wand style for telecom is limited to phone line voltages not 120 volts.) These need to be in close proximity to the wire to use on 120 v wiring & the breaker needs to be turned off. These are less than $100.
The last circuit tracer I use is the cheapest made to find circuit breakers most of these cost ~60$ and require a live circuit to work you plug into an outlet with the transmitter and use the receiver to find the correct breaker.