How to interpret the ban of unions between believers and non-believers?

There is a passage from St Paul's - I can't remember where right now - where he talks about divorce and says that each spouse is redeemed by the other (whichever the faithful one is, the man or the woman) and that is why they should not divorce, but that if the unbeliever wants to leave, the faithful one should let them go. Furthermore, Paul also said the memorable words: "Where there is no love, there is nothing.", so I guess as long as you love your spouse/partner and feel a certain level of attraction for them and they for you, you mustn't worry.

We all have ways of justifying our actions, even when we know deep in our hearts they are neither biblical nor good for us. We can rationalize almost any behavior, as if we're looking for loopholes and exceptions to God's clear commands.

I have a friend (who is not a close friend, but a friend nevertheless) who claims to be a believer in Jesus Christ. Let's call him Carl. In recent years Carl has married a woman whom he knew to be trouble, with a capital T, from the get-go. She and he even had pre-marital counseling in a Christian context. In spite of godly counsel, they committed fornication and she became pregnant. They married, and within a year, they divorced. She went back to drug addiction and Carl was forced to care for the son who was born to them.

Fast forward ten years, and Carl has had a succession of girlfriends, with whom he has committed fornication. In conversations with Carl, I've pointed him to scriptures that prohibit sexual sins, and since these girlfriends are not Christians, I've also warned him about being unequally yoked together with unbelievers. His unbiblical behavior continues.

Carl's latest rationalization is what I call the "missionary relationship" with an unbeliever. In short, Carl believes that his relationship with a woman whom he knows is not a Christian is an opportunity to influence her for Christ. Moreover, in his opinion, she keeps him in line, so to speak, and holds him accountable to a higher standard, knowing he claims to be a Christian.

Is Carl being a missionary in this situation? Of course not! If he is truly a Christian (and I have some doubts about this), will not God hold him responsible for his clear disobedience? Of course He will.

Again, I find it amazing how we all can rationalize our disobedience to God. God, however, sees through our excuse-making, and He wants us to obey Him, but He will not force us. Rather, God wants us to do the right thing, for three reasons: 1) because God says not to, and God knows best; 2) because we love God and want to please Him, even when we find it difficult to do so; and 3) because the world is watching us, and God wants us to be light in a dark world, which is impossible to do when there is willful darkness in our lives.

Sex outside of marriage is strictly prohibited by the Bible. The Bible calls this fornication if you are not married or adultery if you are married but commit it with another person. 1 Corinthians 5 talks about this. It is for our own good that God prohibits this as this cleaves us unto the person that we commit this sin with.

The Bible also tells us not to associate with people like this as friends, as they might influence us. It doesn't tell us to ignore them or hate them, but to love them and pray for them. But do not invite them into your home as they might lead you or your family astray, and especially do not seek a lasting relationship with them (marriage) as this is being unequally yoked.

Be ye not unequally yoked together with unbelievers: for what fellowship hath righteousness with unrighteousness? and what communion hath light with darkness?

2 Corinthians 6:14

EDIT: I'll add a few references to not invite them into you home.

I wrote to you in my epistle not to keep company with sexually immoral people. 10 Yet I certainly did not mean with the sexually immoral people of this world, or with the covetous, or extortioners, or idolaters, since then you would need to go out of the world. 11 But now I have written to you not to keep company with anyone named a brother, who is sexually immoral, or covetous, or an idolater, or a reviler, or a drunkard, or an extortioner—not even to eat with such a person.

1 Corinthians 5:9

It is important to know that this speaks of so-called Christians that are living in open sin.

But sexual immorality and all impurity or covetousness must not even be named among you, as is proper among saints. 4 Let there be no filthiness nor foolish talk nor crude joking, which are out of place, but instead let there be thanksgiving. 5 For you may be sure of this, that everyone who is sexually immoral or impure, or who is covetous (that is, an idolater), has no inheritance in the kingdom of Christ and God. 6 Let no one deceive you with empty words, for because of these things the wrath of God comes upon the sons of disobedience. 7 Therefore do not become partners with them; 8 for at one time you were darkness, but now you are light in the Lord. Walk as children of light 9 (for the fruit of light is found in all that is good and right and true), 10 and try to discern what is pleasing to the Lord. 11 Take no part in the unfruitful works of darkness, but instead expose them.

Ephesians 5:3-11

The example of Lot's family living among the sinful cities of Sodom and Gomorrah should also be noted. Abraham decided to stay in the wilderness while Lot went for the much more attractive valley close to the cities. It was a disaster, as it ruined his home. His family was destroyed. Only He and his two daughters were saved, but it was too late. His daughters were already corrupted with the influence of Sodom and Gomorrah. Because of this, they committed this terrible and vile act of giving their father fermented wine and committing the terrible sin of fornication and incest.