Verse one belongs to the previous chapter (Jesus went unto the Mount of Olives ).
There has been much debate on the authenticity of some of chapter 8. Those with problems with it appear to be enemies of God: such men also usually want to remove other important parts of Scripture! Nevertheless the passage needs to be interpreted with care especially 8:1-11.
1-12 Jesus demonstrates that he came to save not condemn ( katakrino) i.e. He did not come to execute judgement.
13-18 Jesus defends His self witness
19, 20 Jesus answers question, "Where is your father?"
21-30 Jesus tells of His departure
21-36 The outcome of Truth
37-58 The natural man is shown to be wicked, i.e. of the world
59 The Jews seek to kill Jesus
Note: God did not send His Son into the world to condemn the world, but that the world through Him might be saved. John 3:18 (NKJV)
Yes - apparently so as there were witnesses.
Matthew 18: 15-17 deals with witnesses as does the law (Torah eg Deuteronomy 17, 19 and example in 30:19 etc).
Yes, sin has to be identified. A Christian must be able to discern good from evil, to judge right from wrong, see Hebrews 5:14 and 1 Corinthians 2:15
But he who is spiritual judges all things, yet he himself is rightly judged by no one. 1 Corinthians 2:15
Sin in the Church - see 1 Corinthians 5:9-11
Disputes in the church - see 1 Corinthians 6:2-6
The instrument we use is Scripture: For the word of God is living and powerful, and sharper than any two-edged sword, piercing even to the division of soul and spirit, and of joints and marrow, and is a discerner of the thoughts and intents of the heart. Hebrews 4:12 (NKJV). See also Eph 6:10-20
Sin has to be punished - by stoning in this case (See Deuteronomy 5:15, 22:22-29).
Under the law, judgement had to be executed.
Everyone deserves death - hence the judges including the Pharisees, who were under the Law deserved death, and therefore had no place at the judgement seat - they "walked in darkness" see verse 12. This is expanded in Romans 2.
All have sinned and furthermore without the shedding of blood there is no remission!
Grace still means sin has to be punished. Christ was punished in the believer's place, and we are saved if we repent and accept Him as our saviour.
Hence Christ showed that not only could man not keep the law, but the execution of the law in terms of judgement ( krino ) belonged to God only. Salvation hence is by grace alone, Eph 2:8
For by grace are ye saved through faith; and that not of yourselves: [it is] the gift of God:
John displayed Jesus as the Christ who came to save and not condemn.
Condemnation takes place at the great day of Judgement see Rev 20 - call the second death.
All who have rejected Christ will be condemned - and thrown into hell, which was prepared for Satan and his angels.
Hell is everlasting punishment
Matthew 25:46 And these shall go away into everlasting (eternal JND:NT) punishment: but the righteous into life eternal.
 There is no place for universalism here - Christ died for all, but the judgement of sin will still fall on all who do not believed "in the Lord Jesus Christ".