Nine for Mortal Men doomed to die
 

When the Witch King departed from the world, what happened to his Ring? One Ring to find them, One Ring to rule them all springs to mind - did it survive & therefore fall back into the hands of Sauron? Or was this Ring destroyed the moment the Witch King was slain?

The Nine Rings had long been in Sauron's possession. If it wasn't physically destroyed in the fall of Barad-Dur, it was nonetheless shorn of its power when the One was destroyed.

The Rings are probably still in existance, somewhere under the rubble of Barad Dur. Like the three Elven rings, they are now just jewelery.

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If the Ring was still intact, why didn't anyone think of seizing the opportunity to take the Witch King's Ring after his all?

Because it wasn't there. Sauron had taken the Nine Rings back, presumably before the end of the Second Age.

At any rate, if anyone had taken and used the WK's Ring the result would have been exactly what happened to its first owner. An essential point Tolkien is on about is that you *can't* use the Enemy's weapons.

You can take it & use it for evil purposes of your own, i.e. Saruman.

The W-K was destroyed on the Pelennor: Saruman was locked up in Orthanc hundreds of miles away, and Sauron's forces were being routed. Who was there to take it (arguendo), except a Good Guy?

In any event, the W-K's Ring wasn't there: it was safe and sound (for the moment) in Barad-Dur.

Despite a quote in the books that goes something like the Nine the Nazgûl keep, indeed there's plenty of text to support that Sauron took back the Nine Rings. A few examples being...

'True, alas, is our guess. This is not one of the Ulairi, as many have long supposed. It is Sauron himself who has taken shape again and now grows apace; and he is gathering again all the Rings to his hand; and he seeks ever for news of the One, and of the Heirs of Isildur, if they live still on earth.' Of The Rings Of power And The Third Age

'So it is now: the Nine he has gathered to himself; the Seven also, or else they are destroyed.' The Fellowship of the Ring

'They were by far the most powerful of his servants, and the most suitable for such a mission, since they were entirely enslaved to their Nine Rings, which he now himself held; they were quite incapable of acting against his will, ...' Unfinished Tales

Note he now himself held. And...

'... laid upon them by Sauron, who still through their Nine Rings (which he held) had primary control of their wills.' Letters

If that were the case, then the Balrog I guess does not count as a servant of Sauron? Also, what about the Mouth of Sauron - wasn't he on a similar level of power to the Witch King? There is a possibility that, being a lieutenant, he could have been of a higher rank than the Black Captain of Mordor.

There's no suggestion that the Balrog of Moria was serving Sauron or anyone else save itself.

As for the Mouth- he's merely a mortal Man who had learned sorcery. Since Tolkien himself says the Nine are Sauron's most powerful servants, and the W-K the most potent of those, who are we to disagree?

There was a topic here I started not so long time ago, some of the things you mentioned here were discussed there. You may look at the discussion at http://forum.barrowdowns.com/showthread.php?t=13907