In Romans 3:20 Paul writes that no one can be justified through law-keeping.
A contradiction?
No; but Paul is saying that no one can keep the law perfectly; because only doers of the law are justified, AND, no one is justified through law-keeping = no one can keep the law (see also Galatians 6:13).
Paul denied that we can earn our justification as a wage by obeying God's law even if we manage to have perfect obedience to it (Romans 4:1-5), so the reason why we do not earn our justification by obeying it is not because we can't keep it perfectly, but because it was never given as way of earning our justification in the first place, which is why there are many verses that speak against it. However, it remains true that only doers of the law will be justified, so again there must be a reason why our justification requires us to choose to be doers of the law other than in order to earn it as a wage, such as faith insofar as the faith by which we are justified also uphold God's law (Romans 3:28-31).
In Deuteronomy 30:11-20, it says that God's law is not too difficult to obey and that obedience brings life and a blessing while disobedience brings death and a curse, so choose life! So it was presented as a possibility and as a choice, not as the need for perfect obedience.
We are not under the law (as a set of moral tenets) as concerning condemnation.
Would you say that you think that you are under the law in that sense as concerning condemnation?
If so, then I would say that the law condemns you as a sinner and you desperately need the forgiveness of Christ.
It is when we are forgiven that we respond to the Lord in love (Luke 7:36-50) and the righteousness of the law is fulfilled in us (Romans 13:8-10, 8:4).
We are under God's law insofar as we are obligated to obey it. In Romans 8:1, there is no condemnation for those who are Christ because he gave himself to pay the penalty for our sin, however, in 1 John 2:6, those who are in Christ are obligated to walk in the same way he walked, so there is only no condemnation for those who are walking in obedience to God's law.
Sin shall not have dominion over us because we are not under the law but under grace (Romans 6:14).
In Romans 7:7-13, we are taught that freedom from the law means that sin cannot use the law to produce in us all manner of concupiscence; thus causing us to sin.
Grace means that that "wet paint principle" is put in its proper place so that we can walk in freedom and victory over sin.
In Romans 6:14, Paul described the law that are not under as being a law where sin had dominion over us, which does not describe the Law of God, which is a law where holiness, righteousness, and goodness have dominion over us, but rather it is the law of sin where sin had dominion over us. In Romans 6:15, being under grace does not mean that we are permitted to sin, and sin is the transgression of God's law, so we are still under it. In Psalms 119:29, he wanted God to be gracious to him by teaching him to obey His law, so that is what it means to be under grace. We need freedom from the law of sin in order to be free to obey the Law of God, not the other way around.
Romans 7:12 is in the context of the rest of Romans 7. The law being spoken of in the chapter is the same law being spoken of in Romans 7:12.
Romans 7 goes back and forth with contrasting the Law of God with the law of sin. For example, in Romans 7:21-25, Paul said that he delighted in obeying the Law of God and served it with his mind, but contrasted that with the law of sin which held him captive, and which he served with his flesh.
Christ disobeyed the law of God in John 5:16-18.
Sin is the transgression of God's law (1 John 3:4), so to suggest that Jesus disobeyed it is to say that he sinned and therefore to deny that he is our Savior. In John 5:16-18, it states the reasons that the Pharisees had for wanting to kill Jesus, but does not state whether their reasons were correct. It is contradictory to believe both that Jesus is correct about it being lawful to heal on the Sabbath and that the Pharisees were correct about Jesus breaking the Sabbath by healing on it. It has always been lawful to heal on the Sabbath, so the Pharisees were incorrect for thinking that Jesus had broken the Sabbath, but it is nevertheless factual that thinking that Jesus had broken the Sabbath was one of their reasons for why they wanted to kill Jesus
However, His coming was not after a carnal commandment but after the power of an endless life.
There being a change in priesthood, there was also by necessity a change in law.
Christ only disobeyed the letter; not the spirit of what is written.
The set of laws that someone has given paint is a picture of their character. For example, we can see that someone is wise by seeing that they have given wise laws, we can see that they are just by seeing that they have given just laws, and so forth, while a wise person does not given foolish laws, but rather only a fool gives foolish laws. So the Mosaic Law paints us a picture of the character of the God of Israel, and if He had instead commanded things like to commit murder or adultery, then that would have painted a very different picture of His character. If the New Covenant involves following a different set of laws than the Law of Moses, then it involves following a different God with a different set of Character traits than the God of Israel. However, the reality is that the New Covenant still involves following the Mosaic Law (Jeremiah 31:33, Ezekiel 36:26-27), so it is made with the same God.
God's righteousness is eternal (Psalms 119:142), therefore all of His laws for how to testify about His righteousness are also eternal (Psalms 119:160), so Hebrews 7:12 could not be referring to a change of the law in regard to its content, such as with it becoming righteous to commit murder or adultery, but rather the context is speaking about a change of the priesthood, which would also require a change of the law in regard to its administration.
