1) In 431, Our Lady is declared “Mother of God” (Theotokos). The origin of that declaration was the dispute stirred by Nestorius over whether she is the mother of the man Jesus of Nazareth, or mother of God Himself. Cardinal Walter Kasper affirms that Nestorius “found his way even before Chalcedon to the distinction between nature and person [that obviously was not clear all during the 4th century], and anticipated the Chalcedon formula of the one person in two natures. After long being accused in the history of dogma and theology of the gravest heresies, and having even been called a new Judas by the Council of Ephesus, he is now to a large extent being rehabilitated by historical scholarship.” That said, “The controversies occasioned by the conciliar decisions were provoked by Nestorius who, as might be expected from the general character of his Christology, would not speak of Mary as the `mother of God’ (Theotokos) but only as the `mother of Christ’ (Christokos)… The question was whether the Logos is the one subject or whether the unity in Christ… constitutes a tertium quid made up of Godhead and humanity.” Having had the genius to explode the Greek notion of Being (substance) into a distinction of nature and person, Nestorius affirmed that Christ is indeed both God and man, two natures in one Person. However, he failed to see that the relation of our Lady was not merely to the nature of Christ, but as person was related to the Person of Christ – God – as mother (person). Having engendered the human nature, her relation as person-mother is not to nature (“thing”), but to the divine Person of her son (“I”). As Kasper says it: “Because of the identity of the one subject who from eternity is with the Father and who in time has become man, both what is divine and what is human must be predicated of Jesus Christ. Consequently we can and must say that Mary is the mother of God. A second and in practice probably even more important consequence concerned piety, namely, the question whether Jesus’ humanity is to be worshipped. From the fact that the subject is one, it follows that we do not worship Jesus’ humanity as if it were a different subject… but that both are glorified in one single worship….” (Walter Kasper, "Jesus the Christ," Burns and Oates  235). The humanity of Christ is His very Person because the “I” of the Logos of the Father has taken the total humanity of Jesus of Nazareth (no human person) – soul with its faculties of intellect, will, memory, imagination, etc. and body/sentiments – as His own. There is not conflict between the divine and the human because the One Person of the Son has made the total gift of Himself in taking the human as His own. The two wills are the dynamics of the one Person. "Only persons work." Faculties of intellect and will don’t know and will. Persons do.
Therefore, God who is the Creator of the Virgin, takes the humanity of Jesus of Nazareth from her, whereupon she becomes His Mother. The key to this divine maternity is the faith of Mary. The “truth of Mary” is found in these words: “And blessed is she who believed that there would be a fulfillment of what was spoken to her from the Lord" (Lk. 1, 45). Because of her humility and faith, our Lady became the Mother of God: “My mother and my brothers are those who hear the word of God and do it” (Lk. 8, 20-21). Since these words apply to anyone who hears the Word of God and does it, we also can become the Mother of God if we, having heard, do.
2) After the promulgation of this as dogma, Pope Sixtus III consecrated a basilica in Rome in honor of the Blessed Virgin which was later called Saint Mary Major. It is the oldest church dedicated to our Lady.