It seems to me that Gandalf had something of an existential approach to the fate of the Ring and the fellowship. He believed Bilbo (and, by extension, Frodo) was "meant" to have it. He seemed resigned to the notion that it was the hobbits' burden to bear, where elves and men had failed before (and dwarves didn't care). It was Gandalf who foresaw (or guessed) Smeagol's role in the affair and I think he knew the fellowship had to break at some point. This is not to say that Gandalf had actually envisioned all these things, but I think he knew that the very nature of the Ring itself would thwart whatever plans they made, and that he was counting on things he could not see (There is more about them than it would appear).
When I think of Galadriel's motives I am always reminded of the admonition, Do not seek counsel of the elves, for they will say both yes and no. Ultimatey I think they were all counting on the hobbits -- these odd little creatures who were so full of surprises -- to save their bacon. I think they knew the best laid plans of wizards and elves oft go awry, and they were trusting the intrinsic "good" to win out somehow -- even if they could not see the way of it.
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