Most people won't group the next three problems into one, but I will for the sake of keeping this series somewhat short. In 1 Cor. 5-6, Paul addresses the following:
When reading 5:1, you'll think the problem is incest (having inappropriate relations with your stepmother), but upon reading v.2, you'll see that the problem Paul wants to address is how the Corinthian Christians responded to it. Rather than being disgusted by it, they were proud. Paul goes on to share what they should've done - kicked the man out of the church and mourned.
One of the key points in understanding the problem is knowing that the taboo relationship is something that the "pagans" (gentiles) wouldn't even consider (v.1). In other words, while the Church was meant to be the light of the world, the morality in the church has taken such a nose dive that the world outside the Church seems brighter.
The way Paul addresses this by reminding them of their identity. While Christians are meant to love unconditionally on the world as Jesus did, Paul gives them harsh instructions - to excommunicate the man in question. Why would a movement that is a Love Revolution be so rude to one of their own? The explanation is in vv.6-8. A little bit of leaven can change the entire dough.
While Paul is making a subtle reference to the Exodus story, the point is that a little cancer can change (destroy) the entire body. The Corinthian church had forgotten their identity. They were to go and make disciples, but the "cancer" had spread into the congregation and now they can no fulfill their mission.
Put bluntly, Paul is saying, "You are supposed to be the Light of the World. How can you shine your light, when your light is dimmer than the pagans?" Paul is reminding them who they are supposed to be.
The second problem is found in 6:1-11. There's a dispute in the church, and one person (or party) decided to take the other person to court. The courts were known to be completely unfair towards the poor in Corinth. Paul states that there can only be losers in this trial (v.7) because this is proof that they can't fix their own relationships.
The solution Paul offers (in a condescending way) is another reminder of who they are. In v.2, he tells them that they are judges-in-training. Another way of putting it is, according to the Book of Daniel, God's people will one day have authority over the nations. This means, that Christians will one day participate in making wrong things right (oversimplified, I know). His argument is, "What makes you think you'll be fixing the world one day, when you can't even fix this simple relational issue in your church!"
I Can Do Whatever I Want
The final problem (vv.12-20) is a Corinthian saying, "I have the right to do anything" and "Food for the stomach and stomach for the food." These two famous mottos is basically stating, "I can sleep with whoever I want - it's my body!"
Their logic is as follows: If sex was made for my body, then my body was made for sex!
Paul responds to this by reminding them whose they are. He approaches this by a more wholistic line of logic. While he doesn't deny that the body was made for sex, he implies that the body was made for other things as well - eating, reading, walking, working, creating, etc. Therefore, our bodies were made for all these things. All these things are part of God's plan. Therefore, our bodies were made for the Lord.
To top it off, he sets things into a new perspective. If they call themselves Christians, it implies that the Holy Spirit dwells within each of them. When they commit sexual immorality with a prostitute, God is right there with them engaging in the same act. Paul is reminding the Corinthians that they are defiling God.
Leading with the New Identity
There's a big difference between, "You're immature children! Stop suing each other and get along!" and "You are God's people, and God's people will fix the world in the future. Let's start by learning how to fix the relationships in our church!"
One thing I noticed in 1 Corinthians is that Paul never labels the Corinthian Christians as sinners (they deserve to be, especially with so many problems!). He always refers to them as saints. Paul knows the power of leading with the new identity that Jesus had declared for His people.