If (and I do not doubt it) you believe that God is infallible (how could He be otherwise?) and that God's word, the Bible, is inerrant (you are correct but you have to depend upon the traditions of men as translators to say so or upon the Teaching Magisterium which is far more reliable), you are in agreement with Pope St. Pius X's 1907 Syllabus of Errors (Lamentabile Sane) and his brilliant encyclical Pascendi Domenici Gregis against the Modernist heresy (you would call that heresy secular humanism). They are available in translation for cheap prices (less that $5) from the Daughters of St. Paul. All humans save Jesus Christ and Mary are fallen. Most of us, undoubtedly including thee and me and anyone whom we will encounter on FR are fallible. Within the limits stated by other Catholics above, JP II and his predecessors are not fallible when exercising and invoking the infallibility which attaches as a special charism to the papacy through the grace of God. I leave it to those better schooled in that doctrine than I to provide further refinement. am too busy street fighting to reliably indulge in scholarship on the question.
The fallibility includes all churches other than the RCC. I know you find that arrogant. Many found Cassius Marcellus Clay, later Muhammed Ali, to be arrogant. He said he was humble and that the essence of humility was to know one's strengths and weaknesses and freely and honestly confess both. Since Muhammed Ali called himself the greatest, he assumed the burden of proving it. He did prove it and hence he was, by his insightful definition, a humble man, despite the appearances. The RCC is not arrogant for similarly understanding its role and nature in this world in carrying on in obedience to our Founder. If it is, by some definition, arrogant, it has earned the right to that arrogance. In any event it is the only Christian Church enjoying the fullness of the Truth. We draft no one but you are free to enter with appropriately revised beliefs in accordance with the Truth.