All great questions! The easiest way to deal with the repair would be to use an existing wire with a plug on the end, no pause button, and you could get that from another damaged headphone device.

It would probably have three wires in the main cable and then after the splitter, each side would have two wires. Of the three in the main cable, one carries left signal, one carries right signal, and one is ground, aka return; ground is the only wire that would actually split at the splitter. Each headphone would get two wires; one signal, and one return/ground. A pause button complicates the situation; I don't know how those are wired, and as far as I know they work differently for Apple vs Android.

As another person said, these very thin wires are often varnished; each individual strand of wire is covered in a flexible varnish, which insulates it. That varnish needs to be removed before you can solder the wire; I know it is possible but I don't know how. I'd imagine it's done chemically, since the strands are far too thin to withstand any attempt to physically remove the varnish. If I were to try it without finding that info, I'd try burning the varnish off first, then cleaning it very well with a wire brush, and lots of solder flux. Not sure if that would work, but it *seems* like it might. I've never successfully tried anything like this because those wires are so damnedly thin and delicate, I gave up when I tried as a kid.

Are you trying to learn soldering skills, general repair skills, electronics skills, all of the above? Your answers might suggest a more simple starter project.