Some basic facts about the "theory of equations" are at work here.
One (the "factor theorem") is that if f(x) is a polynomial with coefficients in a field (here, the field of rational numbers) and f(t) = 0, then (x  t) is a factor of f(x). This is pertinent to your question (iii).
Also WRT to your question (iii), note that each of the two quantities which are supposed to be roots can be obtained from the other by permuting α and β.
One reason quadratic polynomials are studied (especially as an introduction to the "theory of equations") is that the results are theoretically interesting, and the requisite calculations can actually be done by hand without too much effort. With higherdegree polynomials, the calculations can become too laborious to do by hand, but, armed with a grasp of the theory obtained from the tractable quadratic case, you can at least understand the general form of the results.
One result for which your your question (ii) is a "jumping off point" is known as "Newton's identities."
