I reposted my answer.

I guess many people, including myself, are guilty of making this mistake. At some point I automatically assumed that finite CW-complexes strongly generate $\mathrm{Ho} \mathrm{Top}$, but then I realized that I cannot prove it, which is what led me to the paper I mentioned in the comment. The most striking outcome of those arguments is that both $\mathrm{Ho} \mathrm{Top}$ and $\mathrm{Ho} \mathrm{Top}_*$ fail to have strongly generating sets at all. It just seems that the unstable homotopy category is much weirder than we tend to assume.

I don't know an answer to this question, but I want to point out that contrary to Tyler Lawson's answer linked above Brown's Representability does not hold in the unbased case. A counterexample is discussed in Proposition 2.1 of Heller, Alex On the representability of homotopy functors. J. London Math. Soc. (2) 23 (1981), no. 3, 551–562. Perhaps this example can be also adapted to disprove the based not necessarily connected case, but I'm not so sure about it.

One more thing that may be worth saying explicitly: if $S$ is any class of morphisms and $F : \mathcal{C} \to \mathcal{D}$ a localization with respect to $S$, then $F$ is also a localization with respect to $S_F = \{ f \mid F(f) \textrm{ is an isomorphism} \}$. A class $S$ such that $S = S_F$ is often called saturated.

I'm quite confused. This is indeed obvious, but I assumed that this is not the answer you wanted. After all it does refer to a class of morphisms $S$ with respect to which you localize. Namely $S = \{ f \mid F(f) \textrm{ is an isomorphism} \}$.

I guess so. It also seems that your counterexample should essentially work whenever $\alpha$ is not an epimorphism, so this condition is apparently necessary.