To find four points that a quadratic couldn't possibly fit, remember that quadratics only have one turning point.
To find the quadratic that fits three points, make sure you understand how you can add and subtract graphs, and what happens to the result. Don't try to fit all three points at once – fit two and then “fix” your line to fit the third.
Uniqueness: The Factor Theorem states that if $p$ is a polynomial and $p(a) = 0$, then there is a polynomial $q$ such that $p(x) = (x-a)q(x)$. What does this mean about the degrees of $p$ and $q$?
Finally, to prove two polynomials are equal, try proving their difference is zero.