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) = (x2 - 2)(x2- 2x - 1) the splitting field for BOTH factors is Q(√2) so the galois group is only of order 2, not of order 4.