Yes. A lie can run round the world before the truth has got its boots on.
For the second case, this is why I talk about the nature of effective propaganda. Evidence is the only reliable way to fight against an authority, as the larger group would be significantly less able to censor evidence they are forced to use unverifiable propaganda - this is much less effective for them.
In a limited context, for a limited time, they can. There are many situations where this is all that's needed. (Like, the run up to an election.)
it would be relatively easy for a small group to widely advertise that there is a discrepancy.
A large enough, well established enough group would be able to discredit this discrepancy.