1 Corinthians 9: Is Paul an authority to the Corinthians?

Paul was the missionary who spread the Word to the Corinthians, but the church began to think of Paul as someone with less authority than Peter and other disciples. The church was not financially taking care of Paul and Barnabas. Paul argued that it would be desirable that the church took care of him because he worked for them; however, he continued,  he is working for Jesus, so he wouldn't charge for his service.
What is the purpose of Paul with this discussion? He shows that he cares for the church and not for his money; besides, it is very important that the church understands Paul's authority to value the gospel. 

Who is important to the Corinthian church? Paul wanted that they understood his love for them and mainly his love for Jesus so that they would listen to him. 

1 Corinthians 8: Love is above comfort

In the first century, it was very common that meat bought in the market was a leftover from sacrifices to idols. A believer does not want to have anything to idols but could he eat meat that was sacrificed to idols (a very common situation)? The Corinthians were divided. Some argued that they could eat because meat is just meat and they were not worshiping any idol by eating meat. Paul agreed on that, but he said:

1 Now concerning food sacrificed to idols, we know that we all have knowledge. Knowledge makes one conceited, but love edifies people.

He said that it is better to restrain from eating meat sacrificed to idol for the sake of another Christian that hasn't this understanding. Paul said that love (for a weaker brother) is above the comfort (of eating meat). 
A selfish guy doesn't have the freedom to restrain his comfort for the sake of another person. May we have this freedom.

1 corinthians 7: what is your concern?

 While some members had immoral sexual relations, others, went to another idea: don't have sex even with his spouse. They asked Paul about it. Paul answered in a balanced way: sex in marriage is to please the spouse. Although "no sex" seems much more spiritual, in fact, one's body is not his own; it belongs to the spouse. Paul recommend that in a marriage with a non believer, the Christian must remain the marriage, so that the Christians blesses the non believer.
 The Christian must be guided by love and not by self interest. Sometimes, a great ascetic decision as "no sex" must seem beautiful, but it may be a disguised selfish decision. The Christian must not be concerned about himself but be concerned to praise the Lord.

1 Corinthians 6: called to be rulers

In the Corinthian church, there were some disputes among its members that were taken to Roman tribunals. Paul argues that the Roman law and justice are far inferior to the Kingdom of the Lord rules and justice, besides, Christians are called to rule the world; so how would they rule if they are bellow the law and rules of the world? A ruler must have integrity and this is also seen in how he conducts his sexual life. In the Corinthian world, there was a saying: "Food is for the stomach and the stomach for food"; that is "sex is part of our biology as food is". However, this kind of thought leads one to be ruled by sex when in fact, a Christian is one who finds freedom from himself so that he rules over his own body, which is the Lord's temple.

1 Corinthians 5: Freedom misunderstood

The church of Corinth misunderstood what is grace. A man was having sex with his step mother, sin even to outsiders of the church; however, the church was proud to be inclusive. Paul see the church as arrogant, as boasting about being inclusive when in fact they were not helping the man to be free of his sin. Paul says that the church should expel the man so that he understand his sinner condition and repent.
The Christian church in general doesn't deal with sin very well because the church hardly confronts sin.

1 Corinthians 4: the fight for power

In any organization, it is very common the fight for power: to be friend of who are above and despise who are bellow. The Corinthian church was resembling any human organization when they were discussing who is best: Paul or Apollos? Paul said:

1-4 Don’t imagine us leaders to be something we aren’t. We are servants of Christ, not his masters. We are guides into God’s divine secrets, not security guards posted to protect them. The requirements for a good guide are reliability and accurate knowledge. It matters very little to me what you think of me, even less where I rank in popular opinion. I don’t even rank myself. Comparisons in these matters are pointless. I’m not aware of anything that would disqualify me from being a good guide for you, but that doesn’t mean much. The Master makes that judgment.

A leader is usually afraid of how their subordinates see him, however, if a Christian leader thinks as Paul, then may rest even when he is not well seen because he is aware that he must respond to the Lord and not to men.

1 Corinthians 3: Sinergoi/co-laborers of God.

No all Christians churches are healthy, some are very toxic. Toxic churches are commonly centered in one or more charismatic figures. The church of Corinth were divided, some were more in favor of Paul - the missionary that led many to Christ - and Apollos, the pastor who was with them after Paul left the church. As Paul was the missionary, he could have said: "You are my sons and daughters, never forget this!". However, he said:

9 For we are God’s fellow workers; you are God’s field, God’s building.

He said that he and Apollos was God's fellow workers. They were instruments of the Lord to create the field/church. We all must look to ourselves as sinergoi, co-laborers of God. We work together with God. Sometimes we see ourselves working to God, but instead we must work together with God with a purpose. In this case, working to create a temple built by the believers of Corinth. Would Paul think of himself as the founder of the church that should be respected more than others, he would be working to his own ego.