(I think it's generally better to ask one question per post. I'll take a stab at your first two).
(a) I don't think this equivalence relation is very informative. Over $\mathbb{Q}$, for example, there are lots of degree-5 polynomials with Galois group $S_5$, and they are not related to each other in any particularly illuminating way as far as I can tell. Also, the equivalence relation doesn't respect any algebraic operation, so $F[x]/\sim$ is just a set, which can be described by listing in some way one polynomial for each isomorphism class of Galois groups. This is a computationally hard task - we don't even know what all the Galois groups of $\mathbb{Q}$ are (see "inverse Galois problem").
(b) Consider:
Is your polynomial irreducible over $\mathbb{Q}$? (in your case, degree 3, show that it's irreducible iff it has no rational root. Does it?).
How many real and how many non-real complex roots does it have? (in your case, show that it has exactly one real root).
If there are non-real roots, what can you conclude from "complex conjugation" being an automorphism of the splitting field? (it's an element of order 2 in the Galois group).
Is there an automorphism mapping the real root to one of the non-real ones?
Calculating Galois groups in general is not a straightforward task, but for polynomials of reasonably small degree you can usually get by with considerations such as the above, plus a few more you'll learn with practice.