I found this answer of Grzeszdel in another related thread very much to the point and answers your basic questions in a much more simple manner.
"1) Honorius was a heretic; an ecumenical council has said so, and thus it must be true. The whole tradition of the Catholic Church (rightly considered) understands Honorius to have been anathematized, and anyone who tries to wiggle out of this, however well-intentioned he might be, is actually defending heresy, not Catholicism.
- Popes, speaking in their official capacities as the Petrine successors, have confirmed that Honorius was a heretic (superfluous as this point might be, in light of the conciliar decrees).
- Honorius’ heresy was not merely private, as it was expressed in a letter to one of his brother bishops.
With those point clearly in view, the matter of Honorius and Papal infallibility comes down to this - was Honorius’ (public) heresy expressed in terms which call into question the soundness of chapter IV of Pastor Æternus? Contrary to Fr. Ambrose’s claim, I must insist that the answer to this question is “no.”
Pastor Æternus claims that the Pope “possesses, by the divine assistance promised to him in blessed Peter, that infallibility which the divine Redeemer willed his Church to enjoy in defining doctrine concerning faith or morals” when (and only when) “in the exercise of his office as shepherd and teacher of all Christians, in virtue of his supreme apostolic authority, he defines a doctrine concerning faith or morals to be held by the whole Church.” We have copies of Honorius’ letter to Sergius, the one which earned him his condemnation. It defines nothing, one way or another, and certainly never invokes Honorius’ standing as the Petrine successor. As such, its error is irrelevant to the truth of the claims made in Pastor Æternus.
One is harder pressed to understand Fr. Ambrose’s insistance that Sts. Agatho and Leo think otherwise. Nothing which either of these holy men wrote applies any less well to a man who failed to uphold the truth than it might to a man who affirmatively endorsed falsehood. We all agree that Honorius was a heretic, but it is not obvious to me whence we might conclude that Honorius was some sort of heresiarch."