Did they decide to destroy it simply because they were Wise and knew it was dangerous to wield? Did they create the Fellowship simply because they were heroic; and decided to face Sauron after the Ring was destroyed?
Or did they know that destroying the ring would defeat Sauron, and they wouldn't actually have to do anything once that is taken care of?
Did the Council of Elrond know that destroying the Ring would destroy Sauron?
Yes.
"The Dark Lord Sauron forged in secret a master ring, to control all others. And into this ring he poured his cruelty, his malice, and his will to dominate all life. One Ring to rule them all." —Galadriel regarding Sauron and the forging of the One Ring.
They knew he poured so much of himself into the ring, that destroying it would destroy him with it.
After a brief period of renewed war, Elendil and his people allied with the Elven-king, Gil-Galad to create the Last Alliance, and together fought Sauron. They finally defeated his armies at the Battle of Dagorlad, and sieged Barad-dûr for seven years. Finally, Sauron himself came forth and dueled both Elendil and Gil-galad. He slew them both but Isildur, son of Elendil, took up his father's broken sword, Narsil and cut the One Ring from Sauron's finger, after which Sauron's spirit fled again in ruined form.
The Council of Elrond knew that much of Saurons power came from the One Ring -
Sauron without the Ring lost his ability to take any physical form for a long time, and possibly retreated into the Far East until the time was right.
While Sauron did have a physical form, he remained much weakened without the One Ring and remained hidden in the shadows, directing his armies from afar.
Frodo was able to reach Mount Doom, and upon putting on the Ring, Sauron suddenly became aware of him. Though enraged, he was suddenly gripped with terror, realizing his own folly, and frantically sent the Ringwraiths towards the mountain to retrieve the Ring. He was too late however, and Gollum, after taking the Ring from Frodo, slipped into the Cracks of Doom.
With his source of power destroyed, Sauron was utterly defeated. Barad-dûr fell and his armies were destroyed or scattered, bereft of the driving will behind their conquest, and Mordor itself was shaken to its core as Orodruin belched fire. And, all Eight remaining Nazgul had been in combat with the Great Eagles, and lost, now being helplessly drawn towards the gale of lava on their Fellbeasts, and were consumed.
Gandalf knew; I seem to recall (though I don't have my book with me) that he told Frodo something along the lines of "he poured his greatest power into that ring, and if it is unmade, then his fall should be so low as to... something".
I'll go look for the quote and update if I find it.
Edit: Aha! Wikipedia to the rescue:
Return of the King, ch. 9 The Last Debate:
"If it is destroyed, then he will fall, and his fall will be so low that none can foresee his arising ever again. For he will lose the best part of the strength that was native to him in his beginning, and all that was made or begun with that power will crumble, and he will be maimed for ever, becoming a mere spirit of malice that gnaws itself in the shadows, but cannot again grow or take shape. And so a great evil of this world will be removed."
Yes, they did. At least Elrond knew for it was he (in the movie) who begged Isuldir to throw it into Mt. Doom in the first place. The ring answers to only one master...the one who created it. Not even Gandalf would touch it, and he was more powerful than any present!
To start with, Celebrimbor -- the Noldorian smith who made the Three, the Seven and the Nine -- was very learned in ringlore. (Though not quite so learned as he thought, it seems!) It seems highly likely that he knew what Sauron must have done to create the One and he undoubtedly told the holders of the Three, at least. (When I say "must have done" I'm talking about how a skilled practitioner in a craft can understand how an even more skilled practitioner did something without being able to do it himself.)
In general, in Morgoth's Ring Tolkien writes at length about how Morgoth poured his spirit? essence? being? into Middle Earth at its creation and thus marred it and that Middle Earth was to Morgoth as the One was to Sauron. Some or all of this was known to the Wise either because the Noldor learned it in Valinor before their rebellion, learned it on Middle-Earth from the Valar or Maia during the War of Wrath, or because the Istari brought the knowledge from Valinor. (You'll remember that Saurman was a master of ring lore and while he was probably always rather arrogant, he was not at first an enemy and doubtless shared some of his more general knowledge with the rest of the Wise.)
So: While there's no evidence that any of the Wise at the time of LotR (with the maybe just barely possible exception of Saruman) could make a Ring of Power, they probably had a lot of knowledge about how they worked and what they could do and what they did to those who used them. (Many of us know how to use computers and even how to program computers, but few of us can do chip design.)
Gandalf could thus be certain that the destruction of the Ring would cripple Sauron without being sure it would destroy him. He may only have foreseen as a certainty that the Ring's destruction would put Sauron in the the category of "Defeatable by Mortal Men" -- even only that would have been a big step forward.
(Though it does seem like the Council of Elrond came to a decision rather quickly.)
