I was wondering if we today should apply this law in the Old Testament.
"And if a man entice a maid that is not betrothed, and lie with her, he shall surely endow her to be his wife. If her father utterly refuse to give her unto him, he shall pay money according to the dowry of virgins" (Exodus 22:16-17).
"If a man find a damsel that is a virgin, which is not betrothed, and lay hold on her, and lie with her, and they be found; then the man that lay with her shall give unto the damsel's father fifty shekels of silver, and she shall be his wife; because he hath humbled her, he may not put her away all his days" (Deuteronomy 22:28-29).
It's not talking about rape, rape had another law to it.
Christians do not live under the Old Law, so there is no requirement to follow the rules and regulations found therein. See "Why We Don't Follow the Old Testament" for more details.
However, the Old Testament is still from God and there are many ideas, principles, and examples found in its pages which can teach us much about life. "For whatever things were written before were written for our learning, that we through the patience and comfort of the Scriptures might have hope" (Romans 15:4). Obviously the rule is good and one we ought to seriously consider.
In the past, people did apply this to life. It is what gave rise to the "shotgun weddings" that we heard about in the distant past. Society used to acknowledge that fornication was wrong and when a couple was found to be committing fornication, it was expected by society that the man "do right" by the woman he used.
But for this to happen in our current society, a lot of attitudes toward promiscuous sex would have to change for it to be effective. Notice that even if the man did not marry the woman because her father didn't like him, the man still was required to pay a fine in the form of the dowry price.
We don't use dowries today, and with that we lost the sense that it is a privilege for a man to marry a woman. A silver shekel was worth what an unskilled laborer could earn in a 40 days of work. So the typical dowry price would be roughly what an unskilled laborer could earn in seven years (assuming he had to live on some of his earnings). Think of how attitudes toward marriage would change if society expected a man to work seven years for the privilege of marrying the woman he loved. And think how a young man would think twice before having sex with a young woman if he knew that he would be in debt for seven years if he was caught and possibly forced to marry the young woman with no possibility of a divorce from her.
But the realities of the moment is that our society has made fornication and adultery legal in most cases. Marriage is viewed as a convenience and divorce is available to anyone married for any reason. These immoral attitudes toward sex and marriage make the laws you found impossible to enforce. Even though they are wise laws which effectively discouraged fornication in ancient Israel, they would not work in our current society because of our societies lax attitude toward sin.
However, it does mean that when dealing with two Christians who have committed fornication, I would strongly consider urging the people involved to marry (and I have done so in the past) unless there is something about the situation or people that leads me to believe they are unsuitable marriage partners.