God is not in disagreement with Himself about which laws we should follow, so the Law of God is the same as the Law of Christ, the Law of the Spirit, and the Law of the Father, which was given to Moses. In Deuteronomy 5:31-33, Moses wrote down everything God commanded without departing from it, so all of the Law of Moses was given by God and is thus the Law of God, which is why it is referred to as being the Law of God in verses like Nehemiah 8:1-8, Ezra 7:6-12, and Luke 2:22-23. If Paul had been saying that he was not under the Mosaic Law, then it would have been contradictory for him to say that he was not free from God's law and that he was under Christ's law.
Many of God's laws were given prior to when they were given at Sinai. For example, in Genesis 4:7, God told Cain that sin was crouching at the door and that he must master it, which implies that he already knew what sin is and had already been given laws in that regard. The fact that the penalty for committing murder is death and the fact that Cain was not given the death penalty means that he was not found guilty of committing murder, but rather him being concerned about being avenged and being given protection was in accordance with what would be commanded in Deuteronomy 19 in regard to how to treat someone who commits accidental manslaughter.
In Genesis 7:2, Noah was given instructions for what to do with clean and unclean animals without being told how to tell the difference, and in 8:20, he knew the offer a clean animal, so he must have already been given laws in that regard. In Genesis 39:9, Joseph knew that it was a sin to commit adultery. In Genesis 26:5, Abraham heard God's voice and guarded His charge, commandments, statutes, and laws.
Sin is what is contrary to God's eternal nature and sin is the transgression of God's law because it divides between what is in accordance with or contrary to God's nature. For example, righteousness is in accordance with God's nature while unrighteousness is sin. Sin was in the world because people could act in a way that is contrary to God's nature before they had been given laws again doing that, moreover, the fact that sin was in the world before the law was given means that there were no actions that become righteous or unrighteous when the law was given, but rather the law revealed what has always been and will always be the way to do that.
In Romans 2:13-16, Paul said that only doers of the Mosaic Law will be justified, so he was speaking about believing Gentiles by nature being doers of the Mosaic Law that our consciences bear witness to, not about a separate law. Likewise, in Romans 2:25, the way to recognize that a Gentile has a circumcised heart is by observing their obedience to the Mosaic Law, so he was not switching to speaking about a different law.
David repeatedly said throughout the Psalms that he loved the Mosaic Law and delighted in obeying it, so if we consider the Psalms to be Scripture and to therefore express a correct view of obeying the Mosaic Law, then we will share it as Paul did (Romans 7:22). For example, in Psalms 1-2, blessed are those who delight in the Law of the Lord and who mediate on it day and night, so we can't believe in the truth of these words as Scripture while not allowing them to shape our view of obeying the Mosaic Law.
