At what age should my daughters start wearing a head covering when they pray?

When it used to be common practice to wear veils, girls wore veils to their first communion. I don't recall if they wore veils any younger than that. These days you're free to do as you please, so there is no "right" answer sanctioned by the church any more, if there ever was. You may be able to find a traditionalist Catholic group practicing the old rites who you can ask for a more definitive answer relative to the old practice.

There is currently no requirement in the Catholic Church that women or girls wear a veil. This has technically only been the case since 1983, when the current code of canon law was promulgated.

The 1917 Code of Canon Law did enforce this as a rule:

Viri in ecclesia vel extra ecclesiam, dum sacris ritibus assistunt, nudo capite sint, nisi aliud ferant probati populorum mores aut peculiaria rerum adiuncta; mulieres autem, capite cooperto et modeste vestitae, maxime cum ad mensam Dominicam accedunt.

[Men, either in the church or out of the church, ought to have heads bare while they assist at the sacred rites, unless otherwise justified by established traditions of the people or by particular circumstances. Women, on the other hand, certainly should approach the Lord's table with head covered, and modestly dressed. – my translation]

(1917 Code of Canon Law, Canon 1262, section 2; emphasis added)

The 1983 code, however, abolished this rule along with the rest of the 1917 code of canon law and associated rules:

When this Code takes force, the following are abrogated:

1. the Code of Canon Law promulgated in 1917;

2. other universal or particular laws contrary to the prescripts of this Code unless other provision is expressly made for particular laws;

3. any universal or particular penal laws whatsoever issued by the Apostolic See unless they are contained in this Code;

4. other universal disciplinary laws regarding matter which this Code completely reorders.

(1983 Code of Canon Law, canon 6, section 1)

(Note: you do state that "Vatican canon law also says that later Canon Law abrogates earlier Canon Law only when this is made explicit"; this seems pretty explicit to me.)

If it is not a rule of canon law, is it perhaps addressed by Catholic belief? If so, one would expect to see something about it written in the Catechism, or perhaps addressed by the bishops. I don't see anything to that effect, though. A search through the Catechism reveals no results for "mantilla", and nothing for "veil" as an article of clothing. A quick search through the website of the U.S. Conference of Catholic Bishops for "mantilla" as an article of clothing yielded only a few references, mostly on the Spanish-language pages discussing wedding clothes. I conclude that it is not at all a requirement anymore, certainly not since 1983.

Note: Traditionally, it appears, protocol for papal audiences—particularly for private audiences—has been quite strict and has required among other things that a woman wear a black mantilla; you can see this on Mrs. Obama's head in the picture you post. I cannot find an authoritative statement on whether this is still required as protocol; however, even if it is, it applies simply to visits to the Pope, not to Mass-going in general.

Various cultures and geographic areas—particularly Italy, Spain, and Latin American countries and peoples—have a tradition of wearing a mantilla (chapel veil) independent of any rule of canon law. Since wearing one is certainly not "contrary to the prescripts of this Code" (there's nothing in the current code that says wearing one is wrong), it is an option which Catholic women (and girls) may choose, but which they need not. Under such circumstances it seems understandable for the Church not to legislate an age at which one might consider such a thing, and thus Catholic teaching says nothing at all about the age at which one might begin wearing a chapel veil.

As for tradition and the Bible: certainly Paul does refer to "women" in the passage you cite, but of course he offers no definition for "women". Current canon law sets 18 as the age of majority in the Church (canon 97 sec. 1), 16 as the minimum age at which a person can be affected by penalties in Church law (canon 1323 note 1), 14 as the minimum age at which a woman can marry (canon 1083 section 1—note, dioceses may set a higher minimum), and 7 as the age at which they are responsible for their own acts (canon 97 sec. 2). Any of these might be reasonably considered appropriate to decide whether a young lady is "old enough"; but again there is no official standard.