The nine choirs of angels... I don't know if I can rhyme them off right now: angel, archangel, powers, cherubim, virtues, seraphim, thrones... uhhhhh...
You're missing principalities and dominions. They (with thrones and powers) are mentioned ambiguously by St. Paul, alledgedly. But, again, it's really sort of ambiguous. He could have just meant the worldy powers (ie, secular authorities) given the connotations of those terms.
But StrictCatholicGirl is right about the traditional division and order in Catholic literature. Though, especially early on, you will find many different arrangements of those middle ones, and some even have fewer or greater numbers of choirs.
what else am I missing and what is the purpose for each?
There isnt, strictly speaking, a seperate purpose. It's just a division, rather arbitrary. Some are associated with various things based on their rank. Like the Seraphim with pure charity in adoration for God because they are the closest to him and are portrayed in the Bible with firey wings. The Cherubim with wisdom for various reasons, etc.
Aquinas explains that no two angels are of the same species or even genus (because they have no matter to differentiate them)...but they exist along a continuum from lowest to highest. So, in honor of the Trinity, they are divided equally into 3 hierarchies of 3 choirs each.
how they came to be what they are...
God made all the angels, along a continuum. The exact division may we use may be entirely arbitrary, just pious tradition. But the Bible and stuff does speak of "choirs of angels" and different ones like cherubim, seraphim, and archangel are mentioned...so...it seems like they can be grouped in some general sense.
there are eleven archangels, correct? And, if I also remember correctly they all have ael at the end... even Michael is technically prnounced Mikael. What are all the archangels named? Raphael, Gabriel, Michael... I should know this... what are the others?
Traditionally...Seven actually. Michael, Gabriel, and Raphael are the only ones in the Catholic canon of scripture. But Uriel, Salathiel, Jegudiel, and Barachiel (sometimes Jeremiel) are venerated by the Orthodox and are in their extended canon (ie, apocrypha). This is only somewhat held up by Jewish tradition too.
But "Archangel" in this case (with a capital a) is thought to be different than "archangel" the choir (with a lower-case a).
The Seven Archangels (and thus Michael, Gabriel, and Raphael) are held to be the chief angels. Not in the rather low choir known as archangel (and certainly not comprising the totality of it)...but rather the 7 highest angels, the 7 highest seraphim. Michael at the top (where Lucifer once was), and the 6 surrounding the throne of God that Isaiah saw (and assumed to include Gabriel and Raphael).
But there are not only 7 seraphim, nor just 7 in the choir known as archangel (with a lowercase a). The choirs, according to Aquinas, are simply equal divisions along the continuum. There are likely billions of angels in each choir, with roughly the same number fallen from each choir too among the demons.
I've heard that every angel is paired with a human, as a guardian. And also that we have a specific demon that tempts us particularly. And if the angel's charge is damned...they get assigned another person until someone is saved. So that, in the End, every angel will be particularly paired with someone in heaven (and, it is implied, every demon with someone in hell).It is suggested that this may be the determining factor in what Total Number of angels and humans God decided to make. Namely, that God always intended to make the same number of humans and angels, and then looked forward into these different possible histories, with each possible total number, to see how many of each would fall in each case. And then He chose to create the total number that resulted in a situation with the minimal proportion of fallen, but that also would have the total number of fallen angels and fallen humans be the same. It's a number in the Hundreds of Billions at least, just based on how many humans are thought to have existed so far...
but if the world doesnt end soon, it could be in the trillions or higher, we cant really know. I personally find powerful the idea that the proportion is exactly half of angels and humans fallen, so that every human saved is paired with an angel in heaven, but is also "replacing" a demon, filling up their "spot" in the hierarchy (with Christ replacing Lucifer, etc).
Some say that only the lowest choir known simply as "angels" act as guardians. But I've heard otherwise, I've heard that the higher orders, while also guarding whole nations or groups...do so especially by guarding an important or symbolic member. Like...the guardian angel, let's say a Dominion, of the United States might have also been George Washington's particular personal guardian, etc.
I've also heard it suggested that Michael was Christ's specific guardian in His humanity (and thus the whole Body of Christ in the Church), Gabriel was the Virgin Mary's, and Raphael perhaps St. Joseph's or John the Baptist's (and the other Archangels being assigned to important people too). Of course, it is said that Mary, in addition to her specific guardian, had legions of other angels guarding her. But these likely had their own specific charges too. Perhaps great Saints of the future whose angels were (in God's foresight) honored by also being given this task.