In Catholicism, what is the proper order of the sacraments of initiation and what are the summary arguments in support of this order?

This answer is full of guesses, and it really only concerns the historical reason why the order of reception of the sacraments was re-arranged in the U.S. I leave the canonical explanation and the sacramental explanation to someone with better understanding of these matters than I. I do not expect to receive the bounty offered, but perhaps my answer might help bring a little light into the issue.

Firstly, the order of the reception of the sacraments was not changed universally. In the Eastern and Oriental Catholic Churches (and in the Orthodox churches as well, I believe), all of the sacraments of initiation (Baptism, Confirmation, First Communion) are still administered at once, preferably soon after birth.

The Latin Rite Catholic Church developed a tradition of waiting for confirmation until the bishop could do it himself (CCC 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. The East has kept them united, so that Confirmation is conferred by the priest who baptizes. But he can do so only with the "myron" consecrated by a bishop.

In the Eastern and Oriental Catholic Churches, infant communion is also practiced, so that all sacraments of initiation can be administered at the same time. This is quite valid and licitly allowed in their canon law. In the Latin Rite Church, the close identification of confirmation with the Bishop was desired, and since the Bishop could not be everywhere at once, confirmation was pushed from infancy into early childhood. Because confirmation is the second sacrament of initiation, and First Communion the third, this meant that First Communion was pushed to even later in life. However, the order still remained for the sacraments of initiation: Baptism, Confirmation, then First Communion.

In what follows, I am paraphrasing (hopefully without miss-interpreting) QUAM SINGULARI.

Over time, in the Latin Church there arose a particular tradition on withholding the Eucharist from young children who had not yet attained the age of reason. This arose from piety towards the Eucharist and a desire that those children who received it understood the difference between regular food and Him in the Consecrated Species. This thinking was seen as officially sanctioned by canon 21 of the Fourth Lateran Council which decreed that all faithful were required to confess their sins and receive communion at least once a year after they have reached the age of discernment. (Note that this canon does not actually prohibit the reception of the Eucharist before the age of discernment!).

But the association between age of discernment and the Eucharist had been made. And as time went by many people interpreted the age of discernment as later and later --- some even arguing that full discernment only comes with a mature and complete understanding of the Catholic faith. Thus, the age for First Communion kept getting higher and higher. My guess is that this also caused the age for Confirmation to keep going up, as Confirmation was sandwiched between Baptism and First Communion. But I believe that the order of reception of the sacraments of initiation remained the same: Baptism, Confirmation, First Communion.

At the same time, also in the Latin Rite Church, it started to become a custom to postpone Confirmation after catechesis, thus pushing the Sacrament of Confirmation until after a person could have at least some understanding of the Catholic Faith. This was not at all inconsistent (and in fact, quite consonant) with the reception of First Communion --- the third sacrament of initiation --- thirdly.

So, what happened? Why did the order of reception of these sacraments come to be Baptism, First Communion, and Confirmation in the U.S.?

Pope St. Pius X happened. In Sacra Tridentina he reminded the faithful of how great the Holy Eucharist is, and he encouraged all Catholic people to receive the sacrament often (provided they were properly disposed) --- he even went so far as to suggest daily communion! Soon thereafter Quam Singulari authoritatively asked that The Holy Eucharist not be denied to children past the age of reason --- when they could distinguish the difference between bread and the sacred host --- which the Congregation saw about the age of 7, more or less. Further, the Congregation envisions the reception of First Communion even before the children have finished their catechesis:

A full and perfect knowledge of Christian doctrine is not necessary either for First Confession or for First Communion. Afterwards, however, the child will be obliged to learn gradually the entire Catechism according to his ability.

(These guidelines do seem to require that the age of reason be reached before receiving the Eucharist in the Latin Rite Church, thus forbidding the administering of the Eucharist to very young children and infants). In any case, it seemed that First Communion need not be postponed until after full catechesis has taken place.

My guess is that, in the United States, this was interpreted to mean that First Communion could be (and should be!) received before Confirmation --- because the custom of Confirmation after the end of catechesis was too firmly established to change. And so, in the U.S., the administration of the sacraments (for children) became:

1. The first sacrament of initiation: Baptism.
2. The third sacrament of initiation: First Communion.
3. The second sacrament of initiation: Confirmation.

This new order, however, did not become by any means universal across the entire Catholic Church. As I mentioned earlier, the Eastern and Oriental Catholic churches still administer all of the sacraments of initiation at once, usually during infancy. And furthermore, even in the Latin Rite Church the change in order in the administration of the sacraments did not happen. I know for a fact that in México, circa the middle 1980s (I am not sure about contemporaneous practice), the order of reception for the sacraments of initiation remained the same: Baptism first, Confirmation next, and then First Communion.

Reading Quam Singulari carefully, one notices that there is no requirement therein that the order of the sacraments be altered. Therefore, while its interpretation in the U.S. was to lower the age of First Communion and administer the third sacrament of initiation earlier than the second one, it appears that the interpretation in México (and other places in the word?) --- at least as evidenced by the practice in the mid-1980s --- was to lower the age of Confirmation significantly so that a child could still receive his First Communion at a young age. I suppose that the advantage of the practice in the U.S. is that a child is better catechised by the time Confirmation is administered. The advantage of the practice from México in the mid-1980s is that the sacraments are received in order. The disadvantage of the mid-1980s practice in México is that very often children do not finish their catechesis once they have received all of the sacraments --- catechism is seen by them and their parents these days only as a requirement for the sacraments, not a good and duty in itself ---, so the Catholic education that the faithful have is abysmal. Of course, the disadvantage of the U.S. practice is that the sacraments are received out of order.

As I said before, this answer doesn't touch upon the current canonical or sacramental issues. It makes a stab only at some historical points. A fuller answer would clarify, I think, what the current canon law is on the order of the reception of the sacraments (with perhaps a clarification on the canon law of reception of the Eucharist by infants or by children before the age of discernment), as well as the sacramental implications (if any) of receiving the sacraments out of order.