The way to act in accordance with God's righteousness is straightforwardly based on God's righteousness, not on whether or not someone was a Jew. Likewise, sin was in the world before the Law was given (Romans 5:13), so there were no actions that became righteous or sinful when the Law was given, but rather the Law revealed what has always been and will always be the way to do that. The mark is God's nature and sin is missing the mark, so sin is specific to who God is, not to who Jews are.
The existence of sin requires there to be a standard of what is and is not sin, and this standard is God's Law. Gentiles are either under God's Law and are obligated to obey it, or are not under God's Law, have never needed to refrain from sin, have never needed Jesus to give himself to redeem them from all Lawlessness, and have never needed grace. However, God is sovereign, so we are all under God's Law and obligated to obey it, even those who aren't in a covenant relationship with God, such as with those who were judged in the Flood or when Sodom and Gomorrah were judged for their Lawless deeds. They didn't get the option of whether or not they wanted to be under God's Law and neither do you, but the choice you do get the make is whether or not you are going to heed the Gospel message, repent, and obey.
The verses that you quoted are different ways of saying the same thing, though you did quote 1 John 3:4, which confirms that sin is the transgression of God's law even for Gentiles. God's Law is His instructions for how to do what is holy, righteous, and good, so if you know to do the good that the Law instructs and do not, then that is sin. Likewise, all unrighteousness is sin because it is all in transgression of God's Law. God's Law is His instructions for how to walk in His ways (Deuteronomy 10:12-13), so it distinguishes between what is of the love of the Father and what is of the things of the world.
