[The Saucepan Man]

But... did the Fellowship know that? Because they seemed pretty confident (Gandalf especially) that they would destroy the Ring, if they got to that point.

I doubt that Gandalf knew for sure, but he most probably suspected. I think that he realised from the outset that there was little chance of the Quest succeeding, but he trusted in providence.

I am sure that there is a quote to back that up somewhere, but I think that Gandalf's thoughts concerning the Quest of the Ring were similar to those which he expressed in relation to the Quest of Erebor (from Unfinished Tales):

So it was that that the Quest of Erebor set out. I do not suppose that when it started Thorin had any real hope of destroying Smaug. There was no hope. Yet it happened.

Returning to the topic at hand, I am not so sure that the Fellowship would have remained intact for long, even had Gandalf not fallen in Moria. Even with Gandalf present, one of the Fellowship (most likely Boromir) would have succumbed to the Ring sooner or later and made a play for it. Assuming that he survived, Frodo would have realised that he could not stay with them and made off on his own (no doubt with Sam in tow).

I agree, however, that, had the Fellowship remained intact, the consequences would probably have been disastrous. Most, if not all, would have felt duty bound to accompany Frodo to Mordor, leaving other matters unattended to and (as others have said) making the approach to Mordor far more conspicuous.

But perhaps the most significant consequence of Gandalf not falling in Moria would have been that he would not have returned as the White Wizard, quite possibly leaving him unable to accomplish many of the tasks that he later carried out, such as breaking Saruman's power and challenging the Witch King at the Gates of Minas Tirith.