What is this copper thing in my soldering kit?
Looks like general purpose lacquer-insulated (漆) copper wire. You should be able to solder through the coating without stripping it (it says no need for scraping 不用刮), but I'm not sure if it's sturdy enough (insulation-wise) for use in coils. You can try bending it around and look at it under a microscope or magnifier. 

0.1mm is equivalent to AWG 38, so pretty fine. It might be nice for making dead bug circuits, a bit on the light side for DIPs (AWG 30 is better for that) but okay for fine-pitch SMT parts. 

Pretty handy, really, sometimes enameled wire is very expensive in small lots.