Oops! Missed this thread, 'cause as a Chevy guy I don't venture into "enemy territory" very often.
The "spider" gears are the 4 small ones that Plow Meister mentions, these are what actually provide the "differential" function. In a diagram I'm looking at in a Chevy (sorry!) manual, they are called the "differential pinion" and "differential side gear" - 2 of each - and the cross-shaped piece they are mounted on is called the "differential spider". They will be easy to identify anyway since they are the gears that drive the axle shafts.
I agree with everyone's advice here: A small 110 volt MIG machine will be a little small IMO.
In addition to the fire factor, oil on the parts makes it difficult to get a decent weld. Now, I don't know how much of a chore it is to disassemble the rear end as Pelican01 suggests but I agree it's a good idea if possible. Since it's a spare unit you will be using I'm assuming it's already in the garage.
Regarding preheat and which rod to use, first off a disclaimer: Welding up rear axle gears is something I
haven't done. That said, preheat is a good precaution to take. Good news is, an oven isn't necessary, it can be done with a torch. Yesterday I worked on a bucket gouging off and replacing a tooth shank. Because the cutting edge is hardened it's important to preheat. I used a propane "tiger torch". For the rear axle gears, a regular oxy-propane or oxy-acetylene torch will work fine.
How hot? Without a way of measuring the temperature it's kind of a guessing game, but basically you want to get the "chill" out of the gears. Cutting edges usually are preheated to around 400 F, I would think around 200 F to 300 F should be fine for your gears. This is well below where the metal starts to change colour: red-hot is too hot in this case!
If it feels hot through your welding glove I'd think you're close, 'cause that's how the bucket was yesterday!
The welding part: Since you're using stick, is your machine capable of DC and AC welding current, or AC only? (Most welders in home workshops are AC only) If you have DC capability, use 7018. If it's an AC machine, either 7014 or 6013 will work well. Since you're trying to get into a fairly small area - the gear teeth - I'd suggest using 3/32 diameter rod. Heat setting is usually around 90 to 100 amps for this size rod, but practice on some scrap first.
And once it's welded, one more precaution: Post-heat. This allows the temperature to come down gradually, reducing the chance of the weld or gear itself cracking. Put some more heat on the part right after it's welded, and let it cool down its own -
NO quenching in water!
Let us know how you make out, and post some pics of the burnout competition!