[Legolas]

Good catch on the mortal/invisibility bit, my oversight. It has been a while!

Re: Saruman's task, I meant that the only mission given to the wizards was to take on the forms of old men, travel to Middle-earth, and motivate the races there to combat Sauron. Any other tasks would've been something he chose for himself later on.
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I think any of the Three, specifically, would have been of no use to Saruman in his selfish, evil plan to replace Sauron. As Gandalf stated, the Three (Vilya, Narya, and Nenya) would "endure no evil." Intentionally kept from Sauron by Celebrimbor, they had no evil in them, and so had no tolerance for it. Gandalf's phrase "endure no evil" sounds like they may have even been able to render themselves limp in the hands of evil, basically incomprehensible to one of selfish intent even in the breaking down. We do not know if he had the strength or knowledge to unmake such a great thing. In any event, that is speculation and I think the key is that Saruman did not wish to fight Gandalf head on for it (at least not yet), and was content with the thought that he had Gandalf, along with Narya, imprisoned on his tower.

Saruman needed Gandalf alive because he suspected Gandalf knew the whereabouts of the Ring, which was far more important. Saruman also knew he had very little time; he was in a race against Sauron to find the One. A battle would've been in neither wizard's best interest. Even if Saruman was confident of his odds against Gandalf, he would not have fancied a potential face off with Sauron soon after. Saruman had not yet deceived himself so far as to think he could handle both Gandalf and Sauron.

I imagine the two wizards would have nearly killed each other, much like Gandalf and Durin's Bane (the Moria balrog). Gandalf admits later that he didn't want to fight Saruman as it would've been in vain. I think he'd have said the same of the balrog, only in that matter he had no choice. His situation with Saruman does not reach that point. We do not know what Saruman intended to do with Gandalf apart from pry him for his knowledge of the Ring's location. His approach to Narya could've been as simple as "I'll get it from him tomorrow, after I've had a good breakfast." Saruman's decisions are made under the impression that Gandalf was in his clutches until he decided otherwise; it was not in the plans for him to be flying away on an eagle!
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Still, the wraiths were already invisible by the time Frodo had the Ring:

And they became forever invisible save to him that wore the Ruling Ring, and they entered into the realm of shadows. (Of the Rings of Power and the Third Age, The Silmarillion)

They did not have their rings any longer to use anyway, but if they had, whatever further 'invisibility' you could suggest would be of limited usefulness. Their main power was inciting terror, and they did that with ease. Furthermore, their invisibility did not seem to affect their cloaks or steeds, which they were almost always accompanied by in accounts we're given.

The Nine were subject to counterfeit immortality, a different immortality than Elves; in a way they almost die in that their fundamentally mortal bodies could not endure immortality (this is unchangeable by any save Eru), and so they faded to Wraiths, the invisible creatures Frodo met. They only achieve longevity. After this passing into the shadow, they need not actually have the rings in their possession to continue 'living' or be invisible. Their immortality continued for as long as their rings existed and were powered by the One Ring.

Sauron does indeed hold the Nine Rings, which is how he continued to control them even without the One Ring in his possession. We're told so in a few places:

At length therefore he [Sauron] resolved to use the Ringwraiths. He had been reluctant to do so, until he knew precisely where the Ring was, for several reasons. They were by far the most powerful of his servants, and the most suitable for such a mission, since they were entirely enslaved by their Nine Rings, which he now himself held; they were quite incapable of acting against his will, and if one of them, even the Witch-king their captain, had seized the One Ring, he would have brought it back to his master. (The Hunt for the Ring, Unfinished Tales)

I do not think they could have attacked [Frodo with the One Ring at Weathertop] with violence, nor laid hold upon him or taken him captive; they would have obeyed or feigned to obey any minor command of his that did not interfere with their errand - laid upon them by Sauron, who still through their nine rings (which he held) had primary control of their wills. (#246, Letters of J.R.R. Tolkien)

So it is now; the Nine he had gathered to himself; the Seven also, or else they are destroyed. The Three are hidden still. (Gandalf to Frodo, Fellowship of the Ring)

You saw the Eye of him that holds the Seven and the Nine. (Galadriel to Frodo, Fellowship of the Ring)