Paul addresses a specific incident concerning sexual immorality and tells them how to deal with the person.  “When you are assembled in the name of our Lord Jesus and I am with you in spirit, and the power of our Lord Jesus is present, hand this man over to Satan, so that the sinful nature may be destroyed and his spirit saved on the day of the Lord.”

There are parameters listed in this situation, the group must be spirit-led and assembled together for all the right reasons and within the will of God.  This person (with the sinful nature) has to “hit rock bottom”, be at the bottom of the pit and have no where to go but up. Paul's statements address sin within the Body of Believers. Those within the Body who are pure of heart and mind are obligated to point out the sin so that the person may be convicted of that sin.

Paul warns about the yeast of malice and wickedness (most of the time when yeast is mentioned it is not a good thing, in fact it is a symbol of sin) and encourages them with the bread of sincerity and truth.

“For Christ our Passover lamb has been sacrificed.”
Our sins are forgiven, obliterated, not just covered over temporarily as with the old covenant.

In 1 Corinthians 5 Paul writes "What business is it of mine to judge those outside the church?"  But then in Chapter 6:1-8 Paul writes about the saints judging the world.  This seems to be a contradiction.  In Chapter 5 Paul is saying that he has no authority or jurisdiction over the affairs of the world, but he does have authority among the Body of Believers, just as each of them have that same authority over each other.  They are to evaluate but not condemn.  Concerning judging the world, we as Believers are to evaluate what goes on in the world and are to discern right from wrong. There are several definitions for the word "judge", among those definitions are "to separate, to determine, to evaluate, to produce an opinion concerning right and wrong, to rule, to govern, to judge."  We understand which definition to use based on the context in which it is written, along with what we already know about other Scripture.

Paul lists a number of sins in chapter 6, verses 9-11.  He says that those people committing those sins will not inherit the kingdom of God.  This does not mean that these sinners are not saved. It means they will not inherit rewards.  The parable of the Prodigal Son illustrates this very well.  The prodigal squandered his inheritance.  He was still in the family, but didn't have an inheritance.

Chapter 6:12
Everything is permissible for me - but not everything is beneficial. 

An example of this may be eating certain foods.  Some foods may be delicious but are they really good for us?  Paul says that even though we have the free will we should not let that free will be a license to sin.

Paul separated sexual immorality from other sins. He says that the one who sins sexually sins against his own body and that the body is a temple of the Holy Spirit. Our relationship with Jesus has been repeatedly compared to a marriage relationship. In Corinth as well as in today’s society, sex outside marriage was treated as a normal, even a desirable part of life, while marriage was seen as confining and joyless. Paul made it clear that Christians should take no part in sexual immorality (including homosexual relationships), even if it is accepted in society. It is a sin against God, it hurts all those involved.

“..You are not your own; you were bought with a price…”
Believers don’t belong to themselves.

Concerning marriage and singleness in chapter 7, it appears as if Paul is answering some specific questions concerning specific situations. Some of his answers may seem difficult to understand unless we put a specific question with Paul's answer and we don't know exactly what those questions were. For example when Paul addresses unbelieving spouses, he says, "I, not the Lord say..."  It may be that there were no specific commandments to which Paul could refer and from which he could answer the question.  He had to rely on his judgment based on what the Lord had given him.

Paul expresses that he doesn’t want to be distracted by a family. He believed if he had a family it would slow him down in his ministry.

“But those who marry will face many troubles in this life, and I want to spare you this.”
While we may be tempted to trivialize this statement with "HA!", Paul is merely saying that he wanted to prevent conflict between spouses.

“…but there is one God, the Father…and there is but one Lord, Jesus Christ…”
Paul makes a distinction here between God and Lord, identifying God as the Father and Jesus as the Lord. The word God means supreme deity. The word Lord means supreme authority.

“I have become all things to all men so that by all possible means, I might save some.”
Paul met them where they were without condemning them but he certainly didn't condone their sins. He presented the gospel to the perishing so that some may be saved.

“…for they drank from the spiritual rock that accompanied them, and that rock was Christ.”
No doubt here that it was the Son of God who interacted in the wilderness with Moses.

“So, if you think you are standing firm, be careful that you don’t fall.”
We must stand firm with the whole armor of God, not standing firm with our own confidence (the equivalent of standing on "thin ice").

“And God is faithful; he will not let you be tempted beyond what you can bear…”
Good news for all believers when faced with temptation.

1 Corinthians 10:14
Therefore, my dear friends, flee from idolatry.

Paul writes that we should run away (not just turn away) from anything pagan.  He associates the word pagan with demonic activity.  We should not combine the godly with the pagan.  Nothing pagan can ever be "baptized" into something godly.

In Acts 15:13-21 we read that James decides (and the others agree) to send a letter to the Gentile believers with some guidelines for them to follow.  Among those guidelines were for them to abstain from food sacrificed to idols.  In 1 Corinthians 8:4-6 Paul again brings up food sacrificed to idols. This is in reference to anything pagan, whether it be food, pagan traditions, or any activity designed to honor a false "god". Pagan traditions and pagan holidays do not bring us closer to God.  Honoring His holy days do bring us closer to Him.

Food sacrificed to idols was honoring a false god.  Contrast this with what Jesus said.
John 6:51
I am the living bread that came down from heaven. If anyone eats of this bread, he will live forever. And the bread that I will give for the life of the world is my flesh."

Both Jesus and Paul were speaking about spiritual matters. God continually warns us against bringing anything pagan into our worship of Him.  

1 Corinthians 11:13-14
Now, I want you to realize that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God. Every man who prays or prophesies with his head covered dishonors his head. 

This verse is an example of the Bible interpreting itself.  Paul clearly identified that the head of man is Christ, therefore if the man who prays or prophesies with Christ covered dishonors Christ.  A person who speaks without submission to the truth and authority of the word of Jesus, dishonors Jesus. It's as if that person hides the light of Jesus under a basket. For the remainder of that chapter, it is necessary to put it in the context of the culture of the day.  Women were under the authority of the husband.  Temple prostitutes shaved their heads to identify who they were.  With that cultural mindset, reading the verses again may give a better understanding.