The blackwire problem is where the strands of copper wire are now strands of copper oxide wire. Since there is no actual copper there any more, it is impossible to get it clean enough to solder to. As said before, any thin stranded insulated wire of the right length will do the job. Length is fairly critical for range.
The receiver case will split easily, just needing the clips prying gently. You can then see where the aerial wire is connected - just a simple matter of unsoldering the old, cleaning the hole, poking the tinned end of the new in, soldering down and clipping the box back together. A small tipped, but very hot iron is best - get in and out quick, before the heat gets to where it can do harm.
Insulating the capacitor legs will only help prevent unexpected short circuits due to other bits of metal hitting them. It does nothing to improve interference performance.
It might be a good idea to check out the power wiring at the same time. All sorts of strange things happen when that gets infected, and the only guaranteed cure is replacement.