On the welding rod deal...you don't actually do any welding. If any of you have ever used an arc/stick welder before you will know what I mean when I say stick a welding rod to the shaft. Basically, you just touch the welding rod to the broken off piece and instead of pulling it back, to strike an arc, you hold it on the piece and it melts itself to the piece. So yeah, there may be a quick arc struck, as the rod gets close to the axleshaft, but no more bits of metal or anything will be thrown off the rod than what is likely already in the diff from the broken axleshaft.
fordem....I think that is why they are saying a thin rod...so it can pass between the crosspin and the carrier housing to get through to the broken end of the other shaft.