Does the Catholic Church still stand behind Deuteronomy 22:28-29?

The Catholic Church considers the Bible to be the inspired word of God:

God is the author of Sacred Scripture. "The divinely revealed realities, which are contained and presented in the text of Sacred Scripture, have been written down under the inspiration of the Holy Spirit."

(Catechism of the Catholic Church, paragraph 105; the quote is taken from the Apostolic Constitution Dei Verbum section 11)

The same paragraph indicates that the Church considers the whole of the Old Testament (listed here; note that Deuteronomy is in the list) as canonical and inspired:

Holy Mother Church, relying on the faith of the apostolic age, accepts as sacred and canonical the books of the Old and the New Testaments, whole and entire, with all their parts, on the grounds that, written under the inspiration of the Holy Spirit, they have God as their author and have been handed on as such to the Church herself.

(Dei Verbum section 11, quoted in Catechism paragraph 105)

These inspired books teach the truths necessary for salvation:

The inspired books teach the truth. "Since therefore all that the inspired authors or sacred writers affirm should be regarded as affirmed by the Holy Spirit, we must acknowledge that the books of Scripture firmly, faithfully, and without error teach that truth which God, for the sake of our salvation, wished to see confided to the Sacred Scriptures."

(Catechism paragraph 107; again the quote is Dei Verbum section 11)

The quote you take from Deuteronomy is thus considered inspired and part of canonical Scripture by the Catholic Church. It doesn't necessarily follow that it is giving us instructions which we ought to follow to the letter, but it does follow that there are truths in the book which are needful to salvation.

How, then, should this passage be used? Is it in fact giving us instruction on how to act in our daily lives? Should we be allowing rapists to marry their victims after paying a fine?

Before the reforms implemented by the Second Vatican Council, there were no, or virtually no, readings from the Old Testament prescribed for Sunday Mass—there was one from the epistles or Acts, and one from the Gospels. It was only afterwards that we began reading regularly from the Old Testament, and it was because Catholics need to look at the Old Testament not on its own, but in light of what it can tell us about the New:

...when the Church reads the Old Testament, she searches there for what the Spirit, "who has spoken through the prophets," wants to tell us about Christ.

(Catechism paragraph 702)

We revere the Old Testament because we can find there a wealth of information about God, about Christ—not because we're looking for laws which we must follow. (Certainly there are laws we must follow; the Ten Commandments for example, which follow from the "two greatest commandments" Jesus pointed out. But not all the Old Testament is like that.)

So: what might we be able to get about who and what God is, from these verses, in the light of Christ? There's no official statement from the Church (this is true of most of the Old Testament); but certainly one sees (for example) the importance and the life-long nature of the marriage bond.

As someone else stated in an answer to a similar question of yours, "The Catholic Church does stand behind the verse, but insists that it be read in context, and in the context of the literary genre of the passage." In fact, that would be the answer to any question about whether the Catholic Church "stands behind" a particular verse.

There's a similar law in Exodus 22:15-16, though it deals with seduction rather than rape:

When a man seduces a virgin who is not betrothed, and lies with her, he shall make her his wife by paying the bride price. If her father refuses to give her to him, he must still pay him the bride price for virgins.

Notice that, in the law in Exodus, the obligation is explicitly on the man, and the bride is not the one obligated to go through with the marriage.

It's also there in the law in Deuteronomy that you quoted. Note how it says "he must" and not "she must":

If a man happens to meet a virgin who is not pledged to be married and rapes her and they are discovered, he shall pay her father fifty shekels of silver. He must marry the young woman, for he has violated her. He can never divorce her as long as he lives.

Once he has taken her virginity, which in ancient near eastern culture was even more prized and stigmatized than today, he was entirely beholden to her; if she wishes to marry, he must do so. This verse doesn't explicitly say that she can choose not to marry, but 1) it's there in the similar passage in Exodus, 2) it would only make sense given the verse's emphasis on the obligation of the man to the woman, and 3) there are plenty of rapes recorded in the Bible but no record of the rapist marrying his victim. Therefore it seems logical to conclude that the rape victim has the right to refuse to marry according to this verse in Deuteronomy. That's Fr. Haydock's interpretation as well.

The Catholic Church's position is that the scriptures were authored by God, and that not all of the verses in the Bible, even those that were commands to people in a particular time and place, are meant to be commands to us today. To call any passage in the Bible "erroneous" is to contradict fundamental Catholic doctrine.