Judge not?
Jesus said in Matthew 7:1, "Judge not, that ye be not judged." But then Paul in 1 Corinthians 5:3, "For I verily, as absent in body, but present in spirit, have judged already, as though I were present, concerning him that hath so done this deed,"
This seems like a contradiction, which means there is probably a subtle difference, and thus needs a study.
The word used by Paul in Greek is κρίνω (G2919). This is the exact same word as used by Jesus.
Looking through the usages of the word, we find the following verses:
In Luke 7:43, Jesus tells Simon that he has rightly judged.
In Luke 22:30, Jesus tells the disciples that they will judge the twelve tribes of Israel.
In John, Jesus seems to emphasize righteous judgement, John 7:24, "Judge not according to the appearance, but judge righteous judgment."
John 8:16 suggests that only via the Holy Ghost can right judgements be made, "And yet if I judge, my judgment is true: for I am not alone, but I and the Father that sent me." (As a side note, Jesus clearly talks about Himself here, but Jesus showed us how to live in many ways so perhaps it's also in this case.)
Paulus warns Christians to not partake in unrighteousness and especially judging, Romans 2:1-3, "And thinkest thou this, O man, that judgest them which do such things, and doest the same, that thou shalt escape the judgment of God? But we are sure that the judgment of God is according to truth against them which commit such things. And thinkest thou this, O man, that judgest them which do such things, and doest the same, that thou shalt escape the judgment of God?"
Another clear example of a disciple judging is in Acts 5 where Peter judges both Ananias and Sapphira. He also clearly states that this is to the Holy Ghost, Acts 5:3.
Romans 14 gives a nice example of not judging, Romans 14:3, "Let not him that eateth despise him that eateth not; and let not him which eateth not judge him that eateth: for God hath received him." and Romans 14:10, "But why dost thou judge thy brother? or why dost thou set at nought thy brother? for we shall all stand before the judgment seat of Christ."
Romans 14:12-13, "So then every one of us shall give account of himself to God. Let us not therefore judge one another any more: but judge this rather, that no man put a stumblingblock or an occasion to fall in his brother's way."
That is maybe the core, there is no need to judge since Jesus will do that. For Christians, the focus should probably be on living righteously, spreading the Word, and not to harm anyone else's faith. This makes sense, if you are accidentally falsely criticized in a church, then this can negatively impacts the whole church experience for someone. For teaching, never focus on one person, but always stick to generic arguments and Scripture quotations as Jesus did.
Romans 14:17, "For the kingdom of God is not meat and drink; but righteousness, and peace, and joy in the Holy Ghost."
All in all, judging seems to belong in the same category as aggression. Don't do it yourself, but leave it up to Jesus and God.
James 4:11-12, "Speak not evil one of another, brethren. He that speaketh evil of his brother, and judgeth his brother, speaketh evil of the law, and judgeth the law: but if thou judge the law, thou art not a doer of the law, but a judge. There is one lawgiver, who is able to save and to destroy: who art thou that judgest another?"
Hebrews 10:30, "For we know him that hath said, Vengeance belongeth unto me, I will recompense, saith the Lord. And again, The Lord shall judge his people."
Furthermore, you can ignore other men judging you in food or festive days, Colossians 2:16, "Let no man therefore judge you in meat, or in drink, or in respect of an holyday, or of the new moon, or of the sabbath days:"
At the same time, 1 Corinthians 5:11 tells us to use discernment and in some sense judge, "But now I have written unto you not to keep company, if any man that is called a brother be a fornicator, or covetous, or an idolator, or a railer, or a drunkard, or an extortioner; with such an one no not to eat."
The difference probably lies in the meaning of the word "judge". Need to further investigate...
1 Corinthians 6:1-3, "Dare any of you, having a matter against another, go to law before the unjust, and not before the saints? Do ye not know that the saints shall judge the world? and if the world shall be judged by you, are ye unworthy to judge the smallest matters? Know ye not that we shall judge angels? how much more things that pertain to this life?"
A bit later, 1 Corinthians 6:9, "Know ye not that the unrighteous shall not inherit the kingdom of God?"
Maybe Jesus tells unbelievers to not judge, while Paul instructs Christians how to judge righteously?