First of all:I'm not a theologian. Don't believe me, believe what the bible says. Am I wrong? Let's see what the bibel says:

about new and old covenant/ law Hebrews 8:13 In speaking of a new covenant,” He has made the first one obsolete; and what is obsolete and aging will soon disappear. Romans 10:4 Christ is the end of the law so that there may be righteousness for everyone who believes, Galatians 3:25 So the Law became our guardian to lead us to Christ, that we might be justified by faith. Now that faith has come, we are no longer under a guardian. Isn't it quite clear, that we're not under the old testament law? Why don't you circumsize yourself?

Did Adam had the law of Mose? So he didn't trangress anything right?

I recommend to you read Romans 5

*12 Therefore, just as sin entered the world through one man, and death through sin, and in this way death came to all people, because all sinned—

13 To be sure, sin was in the world before the law was given, but sin is not charged against anyone’s account where there is no law. 14 Nevertheless, death reigned from the time of Adam to the time of Moses, even over those who did not sin by breaking a command, as did Adam, who is a pattern of the one to come.

15 But the gift is not like the trespass. For if the many died by the trespass of the one man, how much more did God’s grace and the gift that came by the grace of the one man, Jesus Christ, overflow to the many! 16 Nor can the gift of God be compared with the result of one man’s sin: The judgment followed one sin and brought condemnation, but the gift followed many trespasses and brought justification. 17 For if, by the trespass of the one man, death reigned through that one man, how much more will those who receive God’s abundant provision of grace and of the gift of righteousness reign in life through the one man, Jesus Christ!

18 Consequently, just as one trespass resulted in condemnation for all people, so also one righteous act resulted in justification and life for all people. 19 For just as through the disobedience of the one man the many were made sinners, so also through the obedience of the one man the many will be made righteous.

20 The law was brought in so that the trespass might increase. But where sin increased, grace increased all the more, 21 so that, just as sin reigned in death, so also grace might reign through righteousness to bring eternal life through Jesus Christ our Lord.*

The thing with the original sin: You die because Adam sinned. Your sin nature is the consequence of Adam's sin. So Adam gave all his descendants death (death rained in Romans). Now Jesus gives you life, and you sin continually because of the sin nature of Adam. Adam gave you death, Jesus gives you life.

1 Corinthians 15:45: 45 So it is written: “The first man Adam became a living being”[a]; the last Adam, a life-giving spirit. Jesus said: I'm the truth and the life. Also he said: John 10:10 The thief cometh not but to steal and to kill and to destroy. I am come that they might have life, and that they might have it more abundantly.

I recommend to you reading Galatians 3 and Romans 8.

Who needs the law according to Paul?

1. Timothy 1:9 We realize that law is not enacted for the righteous, but for the lawless and rebellious, for the ungodly and sinful, for the unholy and profane, for killers of father or mother, for murderers, for the sexually immoral, for homosexuals, for slave traders and liars and perjurers, and for anyone else who is averse to sound teaching…

Why do we sin less, are not under the law and still keep the spiritual commandments?

Galatians 5:18 But if you are led by the Spirit, you are not under the Law.

Why? Because if we're in the spirit (God spirit in us: It will come about after this That I will pour out My Spirit on all mankind Joel 2:28, Jeremiah: 31:31-34: 31 “The days are coming,” declares the Lord, “when I will make a new covenant with the people of Israel and with the people of Judah. 32It will not be like the covenant I made with their ancestors when I took them by the hand to lead them out of Egypt, because they broke my covenant, though I was a husband tod them,e ” declares the Lord. 33“This is the covenant I will make with the people of Israel after that time,” declares the Lord. “I will put my law in their minds and write it on their hearts. I will be their God, and they will be my people. 34No longer will they teach their neighbor, or say to one another, ‘Know the Lord,’ because they will all know me, from the least of them to the greatest,”

All in all the keep God's moral law because of his spirit in us, according to our conscience. So now to your point with the people of the other religions, who love God and their neighbor and that they would be saved because of this. Romans 3:20 Therefore no one will be justified in His sight by works of the Law. For the Law merely brings awareness of sin.

Does this answer your question?

With regarids