Is an unshielded, unbalanced cable useful for anything in hifi? Where?

Well, short runs away from radio stations you don't really need shielding, and the extra capacitance may soften the top end in some cases.

Truthfully, you can get away with unbalanced and unshielded at 20', but it's kind of a crap shoot in regards to radio noise pickup.

Actual shields are only connected at the source, but the ground wire at both.  Same for shielded power cords.

You have much to learn. 

All these things you're talking about, what you need to understand is as good as they sound when talking about them, in the real world you might as well forget about it. It may take you a very long time to realize this, especially with so many chanting technobabble in the echo chamber. But the truth is if you keep at this long enough, and learn to listen closely enough, and pay attention to all the techobabble stories, well then after a good long while what you will find is the stories are good- until they're not. 

What I mean is you will find one thing, and it could be anything. Could be silver. Could be shielded. Could be gold, copper, or silver gold and copper. Could be some super duper geometric weaving of silver, gold, and silver clad copper. Could be nano unobtainium ultra insulatium. Whatever. Does not matter. Point is you hear the story and then you hear the cable and gosh darn it that thing sounds good and so you conclude the story must be true. 

Until who knows how many months or years later you buy something based on the story and its a turd and an expensive turd at that and another one cheaper sounds much better even though dang it doesn't use the storybook magical whatchamacallit. In other words for every really good shielded cable you can find an even better unshielded cable, and on and on and on it goes for XLR, RCA, active, ground plane, you name the story I'll find you the exception that proves that wasn't it after all.  

That is why I must say sorry, but you really must rely on your ears. Because at least those tell the truth. The stories are all just that, stories. There's your real random waste of time and money: stories!   

What you do instead is forget the stories. Stories are for campfires. Focus instead like a laser beam on sound quality. Skim right past the stupid stories and read only the part of the review that talks about how the darn thing sounds. Politely ignore everything the sales man says and when he gets done blathering ask to hear it. Make him plug it in. Or buy stuff like Synergistic that lets you return no questions asked.  

Only way to fly.  

No, shielding is not "needed", but it can be helpful.  Twisting wires provides very effective defense against ambient electrical noise.  Shielding just makes it better, especially at very high frequencies, where twisting becomes less effective.  The best of course is to go balanced, twist it and shield it, but it involves more components in the output stage and that might reduce transparency.  Shielded balanced connection, if properly done, has no shield to wires capacitance, since output is "floating" (not ground related).  My Benchmark AHB2 power amp has only balanced input.  Benchmark stated that 132dB S/N in this amp would be wasted with single ended input.  On the other hand many people use single ended with very high quality gear and achieve great results.