Does the Sacrament of Confirmation go back to the early Church? Also, has it always occurred around the age 14 (except in situations where it would occur later, such as conversions)?
The Sacrament has definitely not always been conferred to those around 14 years of age, but is in reality a rather late development.
To better understand our present practice, it is important to retrace our steps along the path of historical development. We see the first references to the sacrament in the Acts of the Apostles when Peter and John pray that the Holy Spirit comes down upon the Samaritans. Though the Samaritans were baptized they had not yet received the Holy Spirit. As the early Church grew, the sacraments of baptism and confirmation were celebrated in one continuous rite of initiation leading up to the admission and reception of the Holy Eucharist. This is still the current practice in the Eastern Rites of the Church, where the faithful are fully initiated as infants.
After the fifth century, in the west with the principal of the bishop as the celebrant of confirmation, it became difficult for a bishop to travel to the parishes in his diocese to baptize and confirm all at once. Because of this, the separation between baptism, confirmation and the Eucharist grew. Infants were baptized and given communion by the priest and later the bishop would come to administer confirmation. Over time, the infant reception of communion ceased and confirmation received less attention.
In the middle ages, admission to the Eucharist was held off until well after the age of discretion. While confirmation was conferred at the age of discretion, the Eucharist was delayed until the ages of 11 or 12. The order was restored.
Interesting to note is in France, during the mid-1700s, it was decided by a local ordinary that young people be confirmed only after they had received first Eucharist. This was a shift as it was not for the practical reason of the lack of the availability of the bishop but was rather based on adequate instruction. This spread to other dioceses in France. Rome, however, did not approve the practice and Leo XIII in 1897 called for the practice to end and the celebration of confirmation to be at the age of reason.
The displacement of confirmation within the order of Christian initiation was unintentionally begun in 1910 when Pope Pius X lowered the age of first communion to seven. He said nothing of confirmation in his letter, Quam Singulari, but his main concern was that the children have all the resources they need to live a rich spiritual life in order to carry out their mission as Christians in the modern world. Thus, the custom began of receiving First Communion as a 2nd grader and later receiving confirmation in middle or high school. This continues to be a recent practice in the life of the Church.
In the reforms of the Second Vatican Council, the Constitution on the Liturgy called for the rite of confirmation to be revised. Paul VI would clearly state in the Apostolic Constitution on the Sacrament of Confirmation that “The faithful are born anew by baptism, strengthened by the sacrament of confirmation, and finally are sustained by the food of eternal life in the Eucharist. By means of these sacraments of Christian initiation, they thus receive in increasing measure the treasures of divine life and advance towards the perfection of charity.”27 Hence we see the move towards the restoration of the order of the sacraments of initiation: baptism, confirmation and then Eucharist. - “The Sacrament of Confirmation”
According to the Catholic Encyclopedia, the sacrament of Confirmation comes directly from the beginning of the Church, as performed by the Apostles, and thus, it is part of the Oral Revelation (or Tradition, Living Magisterium), and partly seen in the Written Revelation (i.e. the Scriptures). The Encyclopedia states:
The Sacrament of Confirmation is a striking instance of the development of doctrine and ritual in the Church. We can, indeed, detect much more than the mere germs of it in Holy Scripture; but we must not expect to find there an exact description of the ceremony as at present performed, or a complete solution of the various theological questions which have since arisen. It is only from the Fathers and the Schoolmen that we can gather information on these heads.
Then there are two quotes used to support this Biblical basis:
We read in the Acts of the Apostles (8:14-17) that after the Samaritan converts had been baptized by Philip the deacon, the Apostles "sent unto them Peter and John, who, when they were come, prayed for them, that they might receive the Holy Ghost; for he was not yet come upon any of them, but they were only baptized in the name of the Lord Jesus; then they laid their hands upon them, and they received the Holy Ghost".
Again (19:1-6): St. Paul "came to Ephesus, and found certain disciples; and he said to them: Have you received the Holy Ghost since ye believed? But they said to him: We have not so much as heard whether there be a Holy Ghost. And he said: In what then were you baptized? Who said: In John's baptism. Then Paul said: John baptized the people with the baptism of penance . . . Having heard these things, they were baptized in the name of the Lord Jesus. And when Paul had imposed his hands on them, the Holy Ghost came upon them, and they spoke with tongues and prophesied".
From them, the author concludes:
From these two passages we learn that in the earliest ages of the Church there was a rite, distinct from baptism, in which the Holy Ghost was conferred by the imposition of hands (dia tes epitheseos ton cheiron ton Apostolon), and that the power to perform this ceremony was not implied in the power to baptize.
The article goes on to expose evidence and support for the rite later in history, including the Church Fathers.
This perspective is part of the Catholic doctrine, as expressed in the Cathecism (and quoting Paul VI's apostolic constitution Divinae consortium naturae):
1288 "From that time on the apostles, in fulfillment of Christ's will, imparted to the newly baptized by the laying on of hands the gift of the Spirit that completes the grace of Baptism. For this reason in the Letter to the Hebrews the doctrine concerning Baptism and the laying on of hands is listed among the first elements of Christian instruction. the imposition of hands is rightly recognized by the Catholic tradition as the origin of the sacrament of Confirmation, which in a certain way perpetuates the grace of Pentecost in the Church."
Finally, regarding the age of baptism, as stated in the comments by Ken Graham, the Eastern Rites (and the Orthodox Church, in which the sacrament is called Chrismation) performs both sacraments together, whereas the Latin Rite and Western church performs them separately. The same page of the Catechism linked above states:
1290 In the first centuries Confirmation generally comprised one single celebration with Baptism, forming with it a "double sacrament," according to the expression of St. Cyprian. Among other reasons, the multiplication of infant baptisms all through the year, the increase of rural parishes, and the growth of dioceses often prevented the bishop from being present at all baptismal celebrations. In the West the desire to reserve the completion of Baptism to the bishop caused the temporal separation of the two sacraments.
Regarding particularly the Western church, the age at which the confirmation was administered varied considerably. For example, the Encyclopedia states:
It was especially during the thirteenth century that vigorous measures were taken to secure the proper administration of the sacrament. In general, the councils and synods direct the priests to admonish the people regarding the confirmation of their children. The age limit, however, varies considerably. Thus the Synod of Worcester (1240) decreed that parents who neglected to have their child confirmed within a year after birth should be forbidden to enter the church. The Synod of Exeter (1287) enacted that children should be confirmed within three years from birth, otherwise the parents were to fast on bread and water until they complied with the law. At the Synod of Durham (1217? Cf. Wilkins, loc. cit. below) the time was extended to the seventh year.
The latter became, later on, the custom. The Encyclopedia states:
Further, when infant baptism became customary, confirmation was not administered until the child had attained the use of reason. This is the present practice, though there is considerable latitude as to the precise age. The Catechism of the Council of Trent [(1545-1563)] says that the sacrament can be administered to all persons after baptism, but that this is not expedient before the use of reason; and adds that it is most fitting that the sacrament be deferred until the child is seven years old, "for Confirmation has not been instituted as necessary for salvation, but that by virtue thereof we might be found well armed and prepared when called upon to fight for the faith of Christ, and for this kind of conflict no one will consider children, who are still without the use of reason, to be qualified."
This custom of the use of reason being reached at age 7 continues to be in the Canon Law today, under the title "age of discretion", as can be seen here.
This shows that Confirmation was indeed administered at quite different ages across the centuries, and far from the custom of 14 years old, as in some countries today.