1. THE QUESTION WAS: Around what year did the Catholic Church first get its name?
THE ANSWER IS .. A .By 110 AD (more likely 107 A.D.) the church of the Christians was called the Catholic Church in writing (so probably it was used orally before that) by St. Ignatius of Antioch. It has been called the Catholic Church ever since. St. Ignatius, who knew the Apostle St. John personally, was an Early Church Father and a martyr for the Church.
Here are some Church Fathers who wrote about the Catholic Church. From the beginning, the name of this Church was synonymous with being in union with Rome, and with having bishops who were ordained by an apostle, or by a bishop, who by definition, had to be in a direct line from an apostle.
First, we see St. Ignatius of Antioch, a disciple of the apostle St. John, writing in 110 A.D. that the Eucharist is a valid sacrament only when it is celebrated by a bishop of the Catholic Church, or one appointed by a Catholic bishop (a priest):
Let no one do anything of concern to the Church without the bishop. Let that be considered a valid Eucharist which is celebrated by the bishop, or by one whom he appoints. Wherever the bishop appears, let the people be there; just as wherever Jesus Christ is, there is the Catholic Church." (Ignatius of Antioch to the Smyrnaeans 8:1-2 107 A.D..)
THE MARTYRDOM OF POLYCARP
"And of the elect, he was one indeed, the wonderful martyr Polycarp, who in our days was an apostolic and prophetic teacher, bishop of the Catholic Church in Smyrna. For every word which came forth from his mouth was fulfilled and will be fulfilled." (Martyrdom of Polycarp 16:2 [A.D. 155]).
ST. CYRIL OF JERUSALEM
"Inquire not simply where the Lord's house is, for the sects of the profane also make an attempt to call their own dens the houses of the Lord; nor inquire merely where the church is, but where the Catholic Church is. For this is the peculiar name of this Holy Body, the Mother of all, which is the Spouse of Our Lord Jesus Christ" (4th century, Catecheses, xviii, 26).
It is, therefore, the Catholic Church alone which retains true worship. This is the fountain of truth; this, the domicile of faith; this, the temple of God. Whoever does not enter there or whoever does not go out from there, he is a stranger to the hope of life and salvation. . . . Because, however, all the various groups of heretics are confident that they are the Christians and think that theirs is the Catholic Church, let it be known that this is the true Church, in which there is confession and penance and which takes a health-promoting care of the sins and wounds to which the weak flesh is subject. (Lactantius, Divine Institutes 4:30:11-13 [A.D. 307]).
[T]here are many other things which most properly can keep me in [the Catholic Church's] bosom. The unanimity of peoples and nations keeps me here. Her authority, inaugurated in miracles, nourished by hope, augmented by love, and confirmed by her age, keeps me here. The succession of priests, from the very see of the Apostle Peter, to whom the Lord, after his resurrection, gave the charge of feeding his sheep [John 21:15-17], up to the present episcopate, keeps me here. And last, the very name Catholic, which, not without reason, belongs to this Church alone, in the face of so many heretics, so much so that, although all heretics want to be called "Catholic," when a stranger inquires where the Catholic Church meets, none of the heretics would dare to point out his own basilica or house. (Augustine, Against the Letter of Mani Called "The Foundation" 4:5 397 A.D.).
For more info, see http://www.catholic.com/library/What_Catholic_Means.asp
http://www.chnetwork.org/journals/nesschurch/ness_4.htm (What did the name Catholic Church mean in the early Church?
http://ic.net/~erasmus/RAZ12.HTM (The Church)
http://www.turrisfortis.com/hierarchy.html (The Catholic Hierarchy)
http://www.catholicapologetics.info/origin.htm (When Was the Catholic Church founded?)
http://www.chnetwork.org/journals/authority/authority_1.htm (Searching for Authority)
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