I have always thought that The Nine Rings were with The Nazgul, however you raise a good point, I have not got the passage at hand, but I am sure it states it somewhere, I also think at this time of Sauron's spirit fleeing Numenor and taking up the Great Ring again, which implies/means he left it behind when taken prisoner by Ar-Pharazon, or how could his bodiless spirit carry it back from The Downfall, this is not true of The Nazgul as they have a spirit body as such that is knitted together by power, or how else could they cloth themselves in solid matter . I have often wondered if Sauron was tempted to give the three that remained of The Seven to some other persons, would Sarumans greed for power and his jealousy of Gandalf overcome his wisdom, The Three could be worn by a non-elf why not one of the lesser rings. It seems a waste to have them idle, was the Ring that Saruman wore on his finger one of the Seven, we only have Sarumans word that he was a Ring-maker and we know how easily lies came to his mouth. I do not think that the destruction of The Lord of the Nazgul would have resulted in his Ring being destroyed, although once the One was destroyed it would just be a piece of metal. I believe that Legolas shot the winged steed and not the Ringwraith, as an arrow would not do any damage to a Nazgul.
P.S Just read The Council of Elrond, it states that the Nine are with The Nazgul, (The Nine the Nazgul keep states Gandalf).
Good subject Legate
.