last question first:

if all the factors of f(x) have solvable galois groups, then those factors themselves are solvable (by radicals), so f is solvable by radicals.

now let's look at the more general situation:

you are asking: if f(x) = p(x)q(x) in F[x], and if Gal(p) = H, Gal(q) = K, is Gal(f) ≅ HxK?

the thing is, Gal(f) may have a subgroup isomorphic to H, and a subgroup isomorphic to K, but these may "overlap" (even if f is separable).

consider f(x) = (x^{2}- 2)(x^{2}- 2x - 1) the splitting field for BOTH factors is Q(√2) so the galois group is only of order 2, not of order 4.