I hear that Catholic bishops being considered 'infallible' is untrue and their words can be contested if they are not speaking truths directly from the Bible/other relevent documents. However, even if that myth is not entirely so, to what degree are Catholic Bishops opinion's viewed as infallible (if at all)? Is an ordinary Catholic able to gain 'official' undebatable answers even if that answer isn't in the relevent Catholic readings?
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Bishops in themselves are not infallible. However, they do participate in some way in the infallibility of the Catholic Church.
Catholic Bishops participate in the ordinary and universal episcopal magisterium of the Catholic Church. This is a fuzzy set of teachings that are infallible, and come from the authority of the Church to teach The Faith. What exactly are these teachings? They are the teachings on faith and morals that all the Bishops of the world, including the Pope, universally hold as definitive and true. Essentially these are all the true doctrines that all Bishops believe and know to be true, but which haven't been declared to be infallible by a Church Council or the Pope because no one has challenged them in any serious way.
To answer your question somewhat: any Catholic lay person can challenge any Bishop in matters of theology and morals. But most Catholic people will not pay much attention to the challenger because, for the most part, Catholic Bishops are faithful to their calling of teaching the Faith. Thus, there is a presumption that the Bishop as opposed to the challenger is correct, and the challenger would be mostly ignored. In principle, if the challenger happens to be right (and the Bishop wrong), and if the dispute/debate were to gain the attention of the faithful in general and of the majority of Bishops (as well as the Pope), and if the Bishops were to start debating the issue in earnest, some agreeing with the challenger, some with the original Bishop, and if the issue raised and/or the debate surrounding it were causing serious disruptions to the life of Faith, then the Pope (or a Church Council) might be needed to clarify the question. In principle, then, the challenger might actually succeed against the Bishop: not because he is a better debater, but because he was right --- and the fact that his position attracted successful support among the Bishops is the proof of this.
However, for the most part Bishops know the Faith they are teaching. And for the most part Catholic theologians who disagree with a particular Bishop are actually disagreeing with the majority (if not the unanimity) of Catholic Bishops. Therefore their dissent goes nowhere.