The Covenant God made with Noah is still in effect. The "Law" refers to ritual law (temple worship etc.) unless you would argue that the morality of the Law is no longer in force.
Gen 9:12 And God said, This [is] the token of the covenant which I make between me and you and every living creature that [is] with you, for perpetual generations:
Gen 9:13 I do set my bow in the cloud, and it shall be for a token of a covenant between me and the earth.
John 8:7 could be used for anarchy and no consequence for murder, rape, etc. You are missing the entire point of John 8:7.
Verse 6. Tempting him. Trying him, or laying a plan that they might have occasion to accuse him. If he decided the case, they expected to be able to bring an accusation against him; for if he decided that she ought to die, they might accuse him of claiming power which belonged to the Romans--the power of life and death. They might allege that it was not the giving an opinion about an abstract case, but that she was formally before him, that he decided her case judicially, and that without authority or form of trial. If he decided otherwise, they would have alleged that he denied the authority of the law, and that it was his intention to abrogate it. They had had a controversy with him about the authority of the Sabbath, and they perhaps supposed that he would decide this case as he did that--against them. It may be farther added that they knew that Jesus admitted publicans and sinners to eat with him; that one of their charges was that he was friendly to sinners (see Luke 15:2); and they wished, doubtless, to make it appear that he was gluttonous, and a wine-bibber, and a friend of sinners, and disposed to relax all the laws of morality, even in the case of adultery. Seldom was there a plan more artfully laid, and never was more wisdom and knowledge of human nature displayed than in the manner in which it was met.
Wrote on the ground. This took place in the temple. The "ground," here, means the pavement, or the dust on the pavement. By this Jesus showed them clearly that he was not solicitous to pronounce an opinion in the case, and that it was not his wish or intention to intermeddle with the civil affairs of the nation.
As though he heard them not. This is added by the translators. It is not in the original, and should not have been added. There is no intimation in the original, as it seems to be implied by this addition, that the object was to convey the impression that he did not hear them. What was his object is unknown, and conjecture is useless. The most probable reason seems to be that he did not wish to intermeddle; that he designed to show no solicitude to decide the case; and that he did not mean to decide it unless he was constrained to.
Verse 7. They continued asking him. They pressed the question upon him. They were determined to extort an answer from him, and showed a perseverance in evil which has been unhappily often imitated.
Is without sin. That is, without this particular sin; he who has not himself been guilty of this very crime--for in this place the connection evidently demands this meaning.
Let him first cast a stone at her. In the punishment by death, one of the witnesses threw the culprit from the scaffold, and the other threw the first stone, or rolled down a stone to crush him. See Deuteronomy 17:6,7. This was in order that the witness might feel his responsibility in giving evidence, as he was also to be the executioner. Jesus therefore put them to the test. Without pronouncing on her case, he directed them, if any of them were innocent, to perform the office of executioner. This was said, evidently, well knowing their guilt, and well knowing that no one would dare to do it.