the navships wire is not at all suitable for headphone cables IMO, its simply THE MOST microphonic wire I have ever come across, the wire bundle is 26AWG, not the outer coating and the teflon coating on the navships stuff is quite thick, so yes you would need 1/4" nylon, you could possibly get away with 3/8, but I would hesitate, as others have mentioned you need to thread the wire up the sleeve and the friction, even on teflon of a long length is enough to make it a complete PITA.
to stop it from catching when you thread it, put some teflon tape, or heatshrink over the stripped end and ypou wont have this trouble.
the silver wire you mention, I looked it up on ebay and while the price is good, i'm thinking its NOS czech milspec wire, but it also has quite a thick teflon coat and is probably just as microphonic. it is being sold as audio wire, but I doubt it was specified as such when it was made. i'm tempted to grab some to check it out, as stranded pure silver is not often found and the price is good(although a little low in purity; someone will comment on me saying this but I mean in a relative sense) however I would not use it for headphone cables, possibly internal wiring, or for interconnects