If Socrates utters only 'Plato says something false' and Plato only 'Socrates utters something false', then one can be true while the other is false. But which? Perhaps Socrates and Plato have precisely the same intentions and so on when they form the propositions (say each things the other said that God doesn't exist), so that if one is true/false, the other should be true/false. There appears to be nothing to break the symmetry. Buridan's answer is to say that both propositions are false. (This paradox has been called the "no-no" paradox by Roy Sorensen. It is similar to the so-called truth-teller paradox in that the truth-teller is also logically consistent; indeed it can be either true or false, yet there is nothing to break the symmetry.)
For conjunctions, yes; for disjunctions; no. One can certainly know the trivial logical truth that p or that not-p, without knowing which. Presumably for all p, we know that either p or not-p, yet we do not know every truth--we are not omniscient!
He says that, given that the sophism concerns the respondent's response and thus involves self-reference, she should not accept the obligation since you know the proposition to be false (since it involves self-reference).
The proposition is about the future, and if the future is not yet determined, perhaps does not yet even exist, how can the proposition be either true or false? Clearly it cannot be determinately so, but how can it be even indeterminately so? We know that the disjunction, "Either you will throw me in or you won't throw me in" is determinately true even though neither of its disjuncts are, but a lot more would need to be said concerning how the disjuncts have any truth value.