Kevin, Sorry I didn't get to you sooner with this message: >.055" wire . . . It's pretty hard stuff, but shouldn't be harder than the die.
Depends. If it's "piano" or "music" or "spring" wire, and my bet is that it is, it's very hard and you could ruin your carbon steel die and I wouldn't subject my HSS die to it either. Generally any quick cooling, like quenching, will re-hardened a hardenable carbon steel. For these the slower the cooling the better. And for anyone needing to anneal small steel parts . . . . consider silica sand. To heat a small, thin, or delicate part which needs to be evenly heated, and/or heated to exactly a certain color (temp), lay it in a bed of common sand in a shallow metal container and heat it from the bottom. This will heat it slowly and evenly and you can stop the heat at just the right moment. An easy way to slowly cool a small steel item after annealing is to place it (or leave it) in a bed of sand and after it's well soaked with heat cover it with the heated sand and leave it all to cool. It's not a time & temp controlled heat treat oven but it has successfully heat treated many a small bit. Regards, Harry