Hi RINO, welcome to OBOB!
Many of the responses thus far have been thorough, but there are some points I think maybe would be helpful to make in the discussion. I am going to select two of your questions because I think they are of special importance at the present time.
Has Mary always played this large of a role in the Catholic Church?
First, the special in question was just that, a special. Like all specializations, whether in academia or in a television interview, unless one acquires a larger breadth of knowledge, one may be led to believe that "that's that" to a certain topic. It's amazing how common this has become in the soundbite culture of today, in the History Channel generation (my generation).
I did like that one of the people in the special did say, "I believe that only God heals. But I believe also that he can heal through other people." I think that sums up precisely what we Catholics believe in terms of what the Virgin Mary, and all our deceased brothers of sisters (who live in the Lord, interceding for us) do.
The Blessed Virgin has played a huge role in the Catholic Church since the very beginning. Irenaeus in the 2nd century ascribed to her the title of "The New Eve" because she unlocked by her obedience to the angel what Eve locked by her obedience, that is, relationship with God and entrance into heaven.
The furor surrounding the 3rd Ecumenical Council of Ephesus in 431 indicates the fever pitch that Marian Devotion had taken all across the Christian world. Everyone, from the Latin/Germanic West to the Greek East and the Syriac Far East had something to offer in terms of who the Mother of God was, but it was all in terms of Christology. This discussion was universal. The reactions were strong. What this indicates to historians, even beyond the conclusions of the Council itself, is that Marian devotion was already universal and widespread, or else the controversy surrounding the title "Theotokos" would not have given rise to mass riots and civil disorder. In the maxim of Bl. John Henry Newman, no dogma is defined until it is attacked, and in this case, it remains clear that some sort of pious Christian devotion to the Mother of God was in place since Apostolic Christianity. Add to that the findings in archaeology of Churches, statues and mosaics in every corner of the Christian world, and it is clear that she occupies a huge role in the prayer life of Christians in all Christendom.
I make this mention of the universality of the phenomenon because if one believes that Marian devotion arose as a result of the Papacy or some sort of late abberation, one would be mistaken, because it was from the first discernible moments of Christian devotion, devotion to Mary as Mother of Our Savior was a universal phenomenon that transcended linguistic, political and cultural boundaries. Even if one disputes it's theology, it was endemic to Christendom as it emerged from the Catacombs and was permitted to thrive and air out its theological laundry.
How do Catholics sidestep the fact the Bible tells us Jesus had younger siblings including two brothers that became apostles?
Early Christians by and large did not interpret these passages in a manner that would indicate that they believed that Mary had several children. Mind you, these were native speakers of the NT Kione, and many had known, especially in the 2nd century, direct contacts of James, John, and the other apostles. Out of several different hypotheses in the ancient world, at least three out of four, one of whom was skilled in Biblical languages, Jerome, favored the idea that Mary had no other children, and one Greek, Epiphanius, favored the idea that these brethren were Joseph's children from a former marriage.
The only one to support this conclusion that Mary had multiple children is Tertullian, whose cause was taken up by Helvidius, who in turn was fought vigorously by Jerome (and much of Christendom, including Arians). One can see then that the idea of Our Lord having siblings has always been a minority position, and that the vast majority of Christian teachers took for granted that the passages were interpreted otherwise.
Even the touted history of Hegesippus, (how legendary it is another matter) in the 2nd century, incidates a belief among Christians that Joseph had a brother named Clopas, who had James and John as his sons. In this case, Jerome's interpretation of the text as "brothers" referring to the more general Semitic term, meaning, "next of kin" or "relatives" or even "cousins" gains more weight.
I hope this helps.