Part 1

What is the NT definition of "the law?"

If there are 2 "most important" subjects in the NT, understanding what the law means is one of them, because there will be no personal awareness/experience of grace (The New Covenant) without it.

In Romans 10:5, and Galatians 3:12, Paul quotes Leviticus 18:5, and defines the law within this phrase: "The man who does these things will live by them." These texts, along with many others, represent the primary principle of the law you were arranged under (Chapter 1), which is: Receiving from God depends on you. If you do x, God will do y. This concept is not only the foundation of the law for righteousness in the Old Covenant, it's also at the heart of karma, reaping what you sow, and every religion or new age system of thought propagated by sinful human nature around the world. They ALL have the same premise: "The man who does these things will live by them," which is the exact opposite of New Covenant grace.

Deuteronomy 30:15-20 sums up the law by telling Israel if they choose to keep the law, God will bless them; if they don't, they will be cursed. Translation: Whether you experience blessing or cursing in your life depends on you. I.e., The man who does these things will live by them...the law for righteousness.

Therefore, according to the law for righteousness, receiving from God depends on you.